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Get yourself someone who loves you like type theorists love currying

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things i'm interested in separately keep turning out to be related and coming back together in really coherent ways

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List of math things that sound like they're named after multiple guys but actually it's one guy:

Burali-Forti paradox, named after Cesare Burali-Forti

Martin-Löf type theory, named after Per Martin-Löf

Levi-Civita field, named after Tullio Levi-Civita

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I'm interested in paraconsistency, and richer notions of logical negation and things like that. In linear logic they have two of them? Also I read somewhere that the principle of explosion in classical logic is something like the failure to take into a account that positive falsity and negative falsity behave differently. That's cool...

I might need to become a linear logic girlie. Any evangelists out there following this blog?

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I might need to become a linear logic girlie. Any evangelists out there following this blog?

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I do not mean smooth functions! I mean smooth structures, as in maximal smooth atlases. You can restrict smooth structures to open subsets, and if you have two smooth structures on open subsets that agree on their intersection, then they extend to a smooth structure on the union, I think.

But actually, thinking about it more I don't think this presheaf is a sheaf, because it's not separated. If you have an exotic sphere of dimension not equal to 4, and then cover it with open balls, then the restrictions of its smooth structure to these balls must necessarily be diffeomorphic to the standard smooth structure on Euclidean space. So the standard smooth structure and the exotic smooth structure agree on a cover.

Wait smooth structures on open subsets of a topological manifold form a sheaf, right? And a smooth structure on the manifold is then just a global section. Does that work? Can you do cohomology to it?

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oh yeah? well, how does your "set theory" explain this:

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Been learning Toki Pona btw. Really good time so far

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Wait smooth structures on open subsets of a topological manifold form a sheaf, right? And a smooth structure on the manifold is then just a global section. Does that work? Can you do cohomology to it?

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Meine Mama hat mir einfach erlaubt das ich Nullstellen finden darf! Wie cool ist das? Jetzt mach ich Algebra, und trink Cola! Yippie!

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The Topology Game

Here's an updated list of topological properties that I use to play @topoillogical's Topology Game. The rules are as follows. You roll a random integer between -N and N inclusive, where N is the number of properties below (currently N = 114). If the number is negative, you get the negation of the property whose number is the magnitude of the number. You come up with a topological space satisfying this property. This was round 1.

For all future rounds, you start by rolling another property. If the new property follows from the previous properties, prove this. If the new property contradicts the previous properties, prove this. If the new property is independent of all the previous, come up with two spaces satisfying all the previously rolled properties, where one space has the new property and one space has its negation.

You 'win' the game if you manage to solve round 10. You can never truly lose as you can always come back to a given game.

List of 114 properties:

A note on terminology: The space a property refers to is always denoted by X. A neighbourhood of a subset A of X is a subset N of X such that A is contained in the interior of N. Any subset of X is implicitly equipped with the subspace topology.

0. Add your favourite property here to get a slightly higher chance of rolling it.

Property modifiers: These are listed separately, but some specific incarnations (like 'locally connected') are also in the rest of the list because they are more common. If you roll a property modifier, roll again to get the property that is modified. If you roll another modifier, your modifiers stack, so you can get something like 'locally retracts onto a space that contains a compact space'. 1. Locally P: Every point has a neighbourhood basis of P subsets. 2. Somewhere locally P: Some point has a neighbourhood basis of P subsets. 3. Semilocally P: Every point has a P neighbourhood. 4. Somewhere semilocally P: Contains some P subset with non-empty interior. 5. Has a basis of P sets. 6. Hereditarily P: Every subset of X is P. 7. Weakly hereditarily P: Every closed subset of X is P. 8. Contains a P space: X contains some P subset. 9. Retracts onto a P space: X contains a P subset A as a retract, i.e. the inclusion A ↪ X has a continuous left inverse. 10. Is the image of a P space: There is a P space Y and a continuous surjection Y -> X. 11. Homotopy-equivalent to a P space. 12. Weakly homotopy-equivalent to a P space: There is a P space Y and a continuous map X -> Y that induces an isomorphism on all homotopy groups. 13. Is a projective limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical limit. 14. Is an inductive limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical colimit. 15. Is homeomorphic to a space Y: Here Y can be any topological space, for which you will need to find some other way to generate it. Reroll this property if you did not in this round first roll a modifier before.

Classes of spaces 16. Euclidean space. Included for completion, but probably not very interesting to roll. 17. Locally Euclidean. 18. Locally Euclidean with boundary. 19. Homeomorphic to a CW-complex. 20. Homotopy-equivalent to a CW-complex. 21. Homeomorphic to a linearly ordered set with the order topology. 22. Homeomorphic to a topological group. 23. Homeomorphic to a space of continuous maps with the compact-open topology. 24. Spectral: Is homeomorphic to the prime spectrum of a commutative ring with its Zariski topology. 25. Homeomorphic to a vector bundle of positive rank. 26. Homeomorphic to a non-trivial principal bundle.

Separation properties Two subsets A, B are said to be 'separated by P sets' if they have disjoint P neigbourhoods. A 'Urysohn function' for A and B is a function f from X to the real numbers such that A is contained in the preimage of 0 and B is contained in the preimage of 1. A 'perfect Urysohn function' for A and B is a Urysohn function such that A and B equal the preimages of 0 and 1 respectively. If you want you can replace some of the following properties with a weaker version that only holds between topologically distinguishable points, so that your property plus Kolmogorov becomes the property as written. 27. Indiscrete. Included for completion, but probably not very interesting to roll. 28. Point-distinguishing (Kolmogorov): No two points are contained in precisely the same open sets, i.e. any two points are topologically distinguishable. 29. Accessible (Fréchet): Any two points have open neighbourhoods not containing the other point. Equivalently, all singletons are closed. 30. Sober: Any irreducible closed set is the closure of a unique point. 31. Weakly Hausdorff: Any image of a compact Hausdorff space is closed. 32. All compact subsets are closed. 33. Hausdorff: Any two points are separated by open neighbourhoods. 34. Closed Hausdorff: Any two points are separated by closed neighbourhoods. 35. Urysohn Hausdorff: Any two points have a Urysohn function. 36. Perfectly Hausdorff: Any two points have a perfect Urysohn function. 37. Semiregular: The regular open sets, i.e. open sets equal to the interior of their closure, form a base of the topology. 38. Quasiregular: Any non-empty open set contains a non-empty regular open set. 39. Regular: Any point and any closed set not containing it are separated by open neighbourhoods. Not to be confused with being a regular open or closed set. 40. Urysohn regular: Any point and any closed set not containing it have a Urysohn function. 41. Perfectly regular: Any point and any closed set not containing it have a perfect Urysohn function. 42. Normal: Any two disjoint closed sets are separated by open neighbourhoods. Urysohn's lemma states exactly that this is equivalent to Urysohn normal. 43. Hereditarily normal. 44. Fully normal: Any open cover has an open star refinement. 45. Perfectly normal: Any two disjoint closed sets have a perfect Urysohn function. 46. Discrete. Included for completion, but probably not very interesting to roll.

Metrizability properties 47. Metrizable: Is homeomorphic to a metric space. 48. Completely metrizable: Is homeomorphic to a complete metric space. 49. Completely uniformizable: Is homeomorphic to a complete uniform space. Note that 'uniformizable' is equivalent to 'Urysohn regular'. 50. Developable: Admits a countable collection of open covers such that for any closed subset F and point x not in F, one of these covers satisfies that no set in the cover that contains x also intersects F.

Connectedness properties 51. Connected: Has exactly one connected component. 52. Locally connected. 53. Semilocally connected. 54. Path-connected: Any two points are connected by a path. 55. Simple path-connected: Any two distinct points are connected by a simple path, i.e. one that is injective. 56. Locally path-connected. 57. Contractible: Is homotopy-equivalent to a point. 58. Hyperconnected: No two non-empty open subsets are disjoint. 59. Ultraconnected: No two non-empty closed subsets are disjoint. 60. Irreducible: Not the union of two proper closed sets. 61. Dense-in-itself: There are no isolated points. 62. Scattered: Every non-empty subspace has at least one isolated point. 63. Zero-dimensional: The topology has a basis of clopen sets. 64. Totally disconnected: The only connected components are points.

Countability properties 65. Finite: Contains finitely many points. 66. Countable: Contains countably many points. 67. Continuum-sized: Admits a bijection with the real numbers. 68. Supercontinuum-sized: Larger than continuum-sized. 69. First countable: Every point has a countable neighbourhood basis. 70. Second countable: The topology has a countable basis. 71. Separable: Contains a countable dense subset. 72. Sequential: Any set closed under limits of sequences is closed. 73. Baire: If a countable union of closed sets has non-empty interior, then one of the closed sets had non-empty interior. 74. Meagre: Is a countable union of nowhere-dense subsets, i.e. subsets whose closures have empty interior. 75. Gδ space: Every closed set is a countable intersection of open sets. 76. Cosmic: Is the continuous image of a separable metric space.

Compactness properties 77. Compact: Any open cover has a finite subcover. 78. Semilocally compact. Note that this is usually called 'locally compact', but this clashes with my terminology. 79. Lindelöf: Any open cover has a countable subcover. 80. Sequentially compact: Any sequence has a convergent subsequence. 81. Limit point compact: Every infinite subset has a limit point. 82. Pseudocompact: Every real-valued continuous function on X is bounded. 83. σ-compact: Is a countable union of compact subsets. 84. Rim-compact: The topology has a basis of open sets with compact boundaries. 85. Core-compact: For every point x and every neighbourhood U of x there exists a smaller open neighbourhood whose closure in U is compact. 86. Paracompact: Any open cover has a locally finite refinement. 87. Metacompact: Any open cover has a refinement such that every point is contained in only finitely many sets of the refinement. 88. Orthocompact: Any open cover has a refinement such that for every point, the intersection of all open sets in the refinement containing that point is open. 89. Compactly generated: A subset U is open if and only if its intersection with any compact subset K is open in K. 90. Noetherian: Any downward chain of closed sets stabilizes. 91. Admits a partition of unity.

Algebraic-topological properties Several properties to do with homotopy groups require selecting a base point. You can interpret each of these properties as being required to hold at every base point, at some base point, or just requiring that the space is path connected. 92. Simply connected: The fundamental group is trivial. 93. Weakly contractible: All homotopy groups are trivial. 94. Locally contractible. 95. Semilocally semisimply connected: Any point has a neighbourhood N such that the inclusion N ↪ X induces a trivial homomorphism on the fundamental group based at that point. This is usually called 'semilocally simply connected', but this clashes with my terminology. 96. Has an abelian fundamental group. 97. Has a finitely generated fundamental group. 98. Has an uncountable fundamental group.

Miscellaneous properties 99. Alexandrov-discrete: Any intersection of open subsets is open. 100. Door space: Any subset is open or closed or both. 101. Submaximal: Every subset is locally closed, i.e. the intersection of an open set and a closed set. 102. Contains a simple curve: Admits an injective continuous map from the closed interval. 103. Has a cut-point: Contains a non-isolated point x such that if you remove x from its connected component, it is no longer connected. 104. Has a generic point: Contains a dense singleton subset. 105. Homogeneous: For any two points x, y, there is a homeomorphism X -> X carrying x to y. 106. Weakly homogeneous: For any two points x, y and any neighbourhoods U, V of x, y respectively, there exist smaller neighbourhoods U' and V' that are pointedly homeomorphic w.r.t. the base points x, y. 107. Totally heterogeneous: X admits only one homeomorphism X -> X. 108. Uniformly based: Has a basis of homeomorphic subsets. 109. Self-based: Has a basis of subsets homeomorphic to itself. 110. Resolvable: Is the union of disjoint dense subsets. 111. Strongly discrete: Every non-isolated point is an accumulation point of a subset whose points can be simultaneously separated by open neighbourhoods. 112. L-space: Hereditarily Lindelöf and not hereditarily separable. 113. S-space: Hereditarily separable and not hereditarily Lindelöf. 114. Polyadic: Is the continuous image of a power of a one-point compactification of a discrete space.

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Please make it and then show us! :)

The Topology Game

Here's an updated list of topological properties that I use to play @topoillogical's Topology Game. The rules are as follows. You roll a random integer between -N and N inclusive, where N is the number of properties below (currently N = 114). If the number is negative, you get the negation of the property whose number is the magnitude of the number. You come up with a topological space satisfying this property. This was round 1.

For all future rounds, you start by rolling another property. If the new property follows from the previous properties, prove this. If the new property contradicts the previous properties, prove this. If the new property is independent of all the previous, come up with two spaces satisfying all the previously rolled properties, where one space has the new property and one space has its negation.

You 'win' the game if you manage to solve round 10. You can never truly lose as you can always come back to a given game.

List of 114 properties:

A note on terminology: The space a property refers to is always denoted by X. A neighbourhood of a subset A of X is a subset N of X such that A is contained in the interior of N. Any subset of X is implicitly equipped with the subspace topology.

0. Add your favourite property here to get a slightly higher chance of rolling it.

Property modifiers: These are listed separately, but some specific incarnations (like 'locally connected') are also in the rest of the list because they are more common. If you roll a property modifier, roll again to get the property that is modified. If you roll another modifier, your modifiers stack, so you can get something like 'locally retracts onto a space that contains a compact space'. 1. Locally P: Every point has a neighbourhood basis of P subsets. 2. Somewhere locally P: Some point has a neighbourhood basis of P subsets. 3. Semilocally P: Every point has a P neighbourhood. 4. Somewhere semilocally P: Contains some P subset with non-empty interior. 5. Has a basis of P sets. 6. Hereditarily P: Every subset of X is P. 7. Weakly hereditarily P: Every closed subset of X is P. 8. Contains a P space: X contains some P subset. 9. Retracts onto a P space: X contains a P subset A as a retract, i.e. the inclusion A ↪ X has a continuous left inverse. 10. Is the image of a P space: There is a P space Y and a continuous surjection Y -> X. 11. Homotopy-equivalent to a P space. 12. Weakly homotopy-equivalent to a P space: There is a P space Y and a continuous map X -> Y that induces an isomorphism on all homotopy groups. 13. Is a projective limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical limit. 14. Is an inductive limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical colimit. 15. Is homeomorphic to a space Y: Here Y can be any topological space, for which you will need to find some other way to generate it. Reroll this property if you did not in this round first roll a modifier before.

Classes of spaces 16. Euclidean space. Included for completion, but probably not very interesting to roll. 17. Locally Euclidean. 18. Locally Euclidean with boundary. 19. Homeomorphic to a CW-complex. 20. Homotopy-equivalent to a CW-complex. 21. Homeomorphic to a linearly ordered set with the order topology. 22. Homeomorphic to a topological group. 23. Homeomorphic to a space of continuous maps with the compact-open topology. 24. Spectral: Is homeomorphic to the prime spectrum of a commutative ring with its Zariski topology. 25. Homeomorphic to a vector bundle of positive rank. 26. Homeomorphic to a non-trivial principal bundle.

Separation properties Two subsets A, B are said to be 'separated by P sets' if they have disjoint P neigbourhoods. A 'Urysohn function' for A and B is a function f from X to the real numbers such that A is contained in the preimage of 0 and B is contained in the preimage of 1. A 'perfect Urysohn function' for A and B is a Urysohn function such that A and B equal the preimages of 0 and 1 respectively. If you want you can replace some of the following properties with a weaker version that only holds between topologically distinguishable points, so that your property plus Kolmogorov becomes the property as written. 27. Indiscrete. Included for completion, but probably not very interesting to roll. 28. Point-distinguishing (Kolmogorov): No two points are contained in precisely the same open sets, i.e. any two points are topologically distinguishable. 29. Accessible (Fréchet): Any two points have open neighbourhoods not containing the other point. Equivalently, all singletons are closed. 30. Sober: Any irreducible closed set is the closure of a unique point. 31. Weakly Hausdorff: Any image of a compact Hausdorff space is closed. 32. All compact subsets are closed. 33. Hausdorff: Any two points are separated by open neighbourhoods. 34. Closed Hausdorff: Any two points are separated by closed neighbourhoods. 35. Urysohn Hausdorff: Any two points have a Urysohn function. 36. Perfectly Hausdorff: Any two points have a perfect Urysohn function. 37. Semiregular: The regular open sets, i.e. open sets equal to the interior of their closure, form a base of the topology. 38. Quasiregular: Any non-empty open set contains a non-empty regular open set. 39. Regular: Any point and any closed set not containing it are separated by open neighbourhoods. Not to be confused with being a regular open or closed set. 40. Urysohn regular: Any point and any closed set not containing it have a Urysohn function. 41. Perfectly regular: Any point and any closed set not containing it have a perfect Urysohn function. 42. Normal: Any two disjoint closed sets are separated by open neighbourhoods. Urysohn's lemma states exactly that this is equivalent to Urysohn normal. 43. Hereditarily normal. 44. Fully normal: Any open cover has an open star refinement. 45. Perfectly normal: Any two disjoint closed sets have a perfect Urysohn function. 46. Discrete. Included for completion, but probably not very interesting to roll.

Metrizability properties 47. Metrizable: Is homeomorphic to a metric space. 48. Completely metrizable: Is homeomorphic to a complete metric space. 49. Completely uniformizable: Is homeomorphic to a complete uniform space. Note that 'uniformizable' is equivalent to 'Urysohn regular'. 50. Developable: Admits a countable collection of open covers such that for any closed subset F and point x not in F, one of these covers satisfies that no set in the cover that contains x also intersects F.

Connectedness properties 51. Connected: Has exactly one connected component. 52. Locally connected. 53. Semilocally connected. 54. Path-connected: Any two points are connected by a path. 55. Simple path-connected: Any two distinct points are connected by a simple path, i.e. one that is injective. 56. Locally path-connected. 57. Contractible: Is homotopy-equivalent to a point. 58. Hyperconnected: No two non-empty open subsets are disjoint. 59. Ultraconnected: No two non-empty closed subsets are disjoint. 60. Irreducible: Not the union of two proper closed sets. 61. Dense-in-itself: There are no isolated points. 62. Scattered: Every non-empty subspace has at least one isolated point. 63. Zero-dimensional: The topology has a basis of clopen sets. 64. Totally disconnected: The only connected components are points.

Countability properties 65. Finite: Contains finitely many points. 66. Countable: Contains countably many points. 67. Continuum-sized: Admits a bijection with the real numbers. 68. Supercontinuum-sized: Larger than continuum-sized. 69. First countable: Every point has a countable neighbourhood basis. 70. Second countable: The topology has a countable basis. 71. Separable: Contains a countable dense subset. 72. Sequential: Any set closed under limits of sequences is closed. 73. Baire: If a countable union of closed sets has non-empty interior, then one of the closed sets had non-empty interior. 74. Meagre: Is a countable union of nowhere-dense subsets, i.e. subsets whose closures have empty interior. 75. Gδ space: Every closed set is a countable intersection of open sets. 76. Cosmic: Is the continuous image of a separable metric space.

Compactness properties 77. Compact: Any open cover has a finite subcover. 78. Semilocally compact. Note that this is usually called 'locally compact', but this clashes with my terminology. 79. Lindelöf: Any open cover has a countable subcover. 80. Sequentially compact: Any sequence has a convergent subsequence. 81. Limit point compact: Every infinite subset has a limit point. 82. Pseudocompact: Every real-valued continuous function on X is bounded. 83. σ-compact: Is a countable union of compact subsets. 84. Rim-compact: The topology has a basis of open sets with compact boundaries. 85. Core-compact: For every point x and every neighbourhood U of x there exists a smaller open neighbourhood whose closure in U is compact. 86. Paracompact: Any open cover has a locally finite refinement. 87. Metacompact: Any open cover has a refinement such that every point is contained in only finitely many sets of the refinement. 88. Orthocompact: Any open cover has a refinement such that for every point, the intersection of all open sets in the refinement containing that point is open. 89. Compactly generated: A subset U is open if and only if its intersection with any compact subset K is open in K. 90. Noetherian: Any downward chain of closed sets stabilizes. 91. Admits a partition of unity.

Algebraic-topological properties Several properties to do with homotopy groups require selecting a base point. You can interpret each of these properties as being required to hold at every base point, at some base point, or just requiring that the space is path connected. 92. Simply connected: The fundamental group is trivial. 93. Weakly contractible: All homotopy groups are trivial. 94. Locally contractible. 95. Semilocally semisimply connected: Any point has a neighbourhood N such that the inclusion N ↪ X induces a trivial homomorphism on the fundamental group based at that point. This is usually called 'semilocally simply connected', but this clashes with my terminology. 96. Has an abelian fundamental group. 97. Has a finitely generated fundamental group. 98. Has an uncountable fundamental group.

Miscellaneous properties 99. Alexandrov-discrete: Any intersection of open subsets is open. 100. Door space: Any subset is open or closed or both. 101. Submaximal: Every subset is locally closed, i.e. the intersection of an open set and a closed set. 102. Contains a simple curve: Admits an injective continuous map from the closed interval. 103. Has a cut-point: Contains a non-isolated point x such that if you remove x from its connected component, it is no longer connected. 104. Has a generic point: Contains a dense singleton subset. 105. Homogeneous: For any two points x, y, there is a homeomorphism X -> X carrying x to y. 106. Weakly homogeneous: For any two points x, y and any neighbourhoods U, V of x, y respectively, there exist smaller neighbourhoods U' and V' that are pointedly homeomorphic w.r.t. the base points x, y. 107. Totally heterogeneous: X admits only one homeomorphism X -> X. 108. Uniformly based: Has a basis of homeomorphic subsets. 109. Self-based: Has a basis of subsets homeomorphic to itself. 110. Resolvable: Is the union of disjoint dense subsets. 111. Strongly discrete: Every non-isolated point is an accumulation point of a subset whose points can be simultaneously separated by open neighbourhoods. 112. L-space: Hereditarily Lindelöf and not hereditarily separable. 113. S-space: Hereditarily separable and not hereditarily Lindelöf. 114. Polyadic: Is the continuous image of a power of a one-point compactification of a discrete space.

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I was thinking about this yesterday! So in the version of this game in my head it would have a much bigger database of properties obviously, but also many more properties that themselves take arguments (dependent properties, if you will). For example, the property of 'being a K(G,n)' (which for the uninitiated means being an Eilenberg-MacLane space of a proscribed type) takes a group G (which is sometimes denoted by π, like @positively-knotted did) and a natural number n as arguments.

To make that work you'd need a random variable that spits out an arbitrary natural number (which to keep things interesting you would want to have a pretty fat tail, I was thinking something like P(N = n) = 1/n(n+1)), but more pressingly you'd also need to generate a random group G. To keep things interesting, you'd also want this group to be abelian if n > 1, because otherwise you have a trivial contradiction by the Eckmann-Hilton argument.

To support this feature, the game would also need a database of different groups up to isomorphism, as well as a list of group-theoretic properties. At that point you might as well open up the game to be played with groups instead of topological spaces.

This is part of the reason I included property 15 above ("Is homeomorphic to a space Y"), which I believed led to some confusion for @lorxus-is-a-fox in their game. Here, much like the group G, you would need to randomly generate a specific topological space (up to homeomorphism), which I was imagining you could do by just throwing a dart at the Wikipedia List of topologies. In the more advanced game, property 15 would have a 0% chance of being rolled for the first roll of a round, but a much higher chance on subsequent rolls after the first roll was a dependent property. You could get a property like 'every point has an open neighbourhood homeomorphic to the real projective plane', which I thought was interesting.

Actually, looking at that Wikipedia list it cites the website π-Base, which seemingly has exactly the database I envisioned, complete with searchable properties and counterexamples. Huh!

Anyway, you can get much crazier than this. Get a database of properties of any kind of mathematical structure, and then keep track of the functors between these structures. Now your first roll will be to select a category. If you get 'metric spaces', you get access to rolling metric properties like convexity or boundedness or completeness, but also through the forgetful functor to all the topological properties above, or all set-theoretic properties. Non-forgetful functors are also available; you can roll group properties for its homotopy groups, lattice properties for its lattice of closed subsets, or its (unbounded) lattice of complete subsets, or chain complex properties of its Čech nerve, whatever you like!

The Topology Game

Here's an updated list of topological properties that I use to play @topoillogical's Topology Game. The rules are as follows. You roll a random integer between -N and N inclusive, where N is the number of properties below (currently N = 114). If the number is negative, you get the negation of the property whose number is the magnitude of the number. You come up with a topological space satisfying this property. This was round 1.

For all future rounds, you start by rolling another property. If the new property follows from the previous properties, prove this. If the new property contradicts the previous properties, prove this. If the new property is independent of all the previous, come up with two spaces satisfying all the previously rolled properties, where one space has the new property and one space has its negation.

You 'win' the game if you manage to solve round 10. You can never truly lose as you can always come back to a given game.

List of 114 properties:

A note on terminology: The space a property refers to is always denoted by X. A neighbourhood of a subset A of X is a subset N of X such that A is contained in the interior of N. Any subset of X is implicitly equipped with the subspace topology.

0. Add your favourite property here to get a slightly higher chance of rolling it.

Property modifiers: These are listed separately, but some specific incarnations (like 'locally connected') are also in the rest of the list because they are more common. If you roll a property modifier, roll again to get the property that is modified. If you roll another modifier, your modifiers stack, so you can get something like 'locally retracts onto a space that contains a compact space'. 1. Locally P: Every point has a neighbourhood basis of P subsets. 2. Somewhere locally P: Some point has a neighbourhood basis of P subsets. 3. Semilocally P: Every point has a P neighbourhood. 4. Somewhere semilocally P: Contains some P subset with non-empty interior. 5. Has a basis of P sets. 6. Hereditarily P: Every subset of X is P. 7. Weakly hereditarily P: Every closed subset of X is P. 8. Contains a P space: X contains some P subset. 9. Retracts onto a P space: X contains a P subset A as a retract, i.e. the inclusion A ↪ X has a continuous left inverse. 10. Is the image of a P space: There is a P space Y and a continuous surjection Y -> X. 11. Homotopy-equivalent to a P space. 12. Weakly homotopy-equivalent to a P space: There is a P space Y and a continuous map X -> Y that induces an isomorphism on all homotopy groups. 13. Is a projective limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical limit. 14. Is an inductive limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical colimit. 15. Is homeomorphic to a space Y: Here Y can be any topological space, for which you will need to find some other way to generate it. Reroll this property if you did not in this round first roll a modifier before.

Classes of spaces 16. Euclidean space. Included for completion, but probably not very interesting to roll. 17. Locally Euclidean. 18. Locally Euclidean with boundary. 19. Homeomorphic to a CW-complex. 20. Homotopy-equivalent to a CW-complex. 21. Homeomorphic to a linearly ordered set with the order topology. 22. Homeomorphic to a topological group. 23. Homeomorphic to a space of continuous maps with the compact-open topology. 24. Spectral: Is homeomorphic to the prime spectrum of a commutative ring with its Zariski topology. 25. Homeomorphic to a vector bundle of positive rank. 26. Homeomorphic to a non-trivial principal bundle.

Separation properties Two subsets A, B are said to be 'separated by P sets' if they have disjoint P neigbourhoods. A 'Urysohn function' for A and B is a function f from X to the real numbers such that A is contained in the preimage of 0 and B is contained in the preimage of 1. A 'perfect Urysohn function' for A and B is a Urysohn function such that A and B equal the preimages of 0 and 1 respectively. If you want you can replace some of the following properties with a weaker version that only holds between topologically distinguishable points, so that your property plus Kolmogorov becomes the property as written. 27. Indiscrete. Included for completion, but probably not very interesting to roll. 28. Point-distinguishing (Kolmogorov): No two points are contained in precisely the same open sets, i.e. any two points are topologically distinguishable. 29. Accessible (Fréchet): Any two points have open neighbourhoods not containing the other point. Equivalently, all singletons are closed. 30. Sober: Any irreducible closed set is the closure of a unique point. 31. Weakly Hausdorff: Any image of a compact Hausdorff space is closed. 32. All compact subsets are closed. 33. Hausdorff: Any two points are separated by open neighbourhoods. 34. Closed Hausdorff: Any two points are separated by closed neighbourhoods. 35. Urysohn Hausdorff: Any two points have a Urysohn function. 36. Perfectly Hausdorff: Any two points have a perfect Urysohn function. 37. Semiregular: The regular open sets, i.e. open sets equal to the interior of their closure, form a base of the topology. 38. Quasiregular: Any non-empty open set contains a non-empty regular open set. 39. Regular: Any point and any closed set not containing it are separated by open neighbourhoods. Not to be confused with being a regular open or closed set. 40. Urysohn regular: Any point and any closed set not containing it have a Urysohn function. 41. Perfectly regular: Any point and any closed set not containing it have a perfect Urysohn function. 42. Normal: Any two disjoint closed sets are separated by open neighbourhoods. Urysohn's lemma states exactly that this is equivalent to Urysohn normal. 43. Hereditarily normal. 44. Fully normal: Any open cover has an open star refinement. 45. Perfectly normal: Any two disjoint closed sets have a perfect Urysohn function. 46. Discrete. Included for completion, but probably not very interesting to roll.

Metrizability properties 47. Metrizable: Is homeomorphic to a metric space. 48. Completely metrizable: Is homeomorphic to a complete metric space. 49. Completely uniformizable: Is homeomorphic to a complete uniform space. Note that 'uniformizable' is equivalent to 'Urysohn regular'. 50. Developable: Admits a countable collection of open covers such that for any closed subset F and point x not in F, one of these covers satisfies that no set in the cover that contains x also intersects F.

Connectedness properties 51. Connected: Has exactly one connected component. 52. Locally connected. 53. Semilocally connected. 54. Path-connected: Any two points are connected by a path. 55. Simple path-connected: Any two distinct points are connected by a simple path, i.e. one that is injective. 56. Locally path-connected. 57. Contractible: Is homotopy-equivalent to a point. 58. Hyperconnected: No two non-empty open subsets are disjoint. 59. Ultraconnected: No two non-empty closed subsets are disjoint. 60. Irreducible: Not the union of two proper closed sets. 61. Dense-in-itself: There are no isolated points. 62. Scattered: Every non-empty subspace has at least one isolated point. 63. Zero-dimensional: The topology has a basis of clopen sets. 64. Totally disconnected: The only connected components are points.

Countability properties 65. Finite: Contains finitely many points. 66. Countable: Contains countably many points. 67. Continuum-sized: Admits a bijection with the real numbers. 68. Supercontinuum-sized: Larger than continuum-sized. 69. First countable: Every point has a countable neighbourhood basis. 70. Second countable: The topology has a countable basis. 71. Separable: Contains a countable dense subset. 72. Sequential: Any set closed under limits of sequences is closed. 73. Baire: If a countable union of closed sets has non-empty interior, then one of the closed sets had non-empty interior. 74. Meagre: Is a countable union of nowhere-dense subsets, i.e. subsets whose closures have empty interior. 75. Gδ space: Every closed set is a countable intersection of open sets. 76. Cosmic: Is the continuous image of a separable metric space.

Compactness properties 77. Compact: Any open cover has a finite subcover. 78. Semilocally compact. Note that this is usually called 'locally compact', but this clashes with my terminology. 79. Lindelöf: Any open cover has a countable subcover. 80. Sequentially compact: Any sequence has a convergent subsequence. 81. Limit point compact: Every infinite subset has a limit point. 82. Pseudocompact: Every real-valued continuous function on X is bounded. 83. σ-compact: Is a countable union of compact subsets. 84. Rim-compact: The topology has a basis of open sets with compact boundaries. 85. Core-compact: For every point x and every neighbourhood U of x there exists a smaller open neighbourhood whose closure in U is compact. 86. Paracompact: Any open cover has a locally finite refinement. 87. Metacompact: Any open cover has a refinement such that every point is contained in only finitely many sets of the refinement. 88. Orthocompact: Any open cover has a refinement such that for every point, the intersection of all open sets in the refinement containing that point is open. 89. Compactly generated: A subset U is open if and only if its intersection with any compact subset K is open in K. 90. Noetherian: Any downward chain of closed sets stabilizes. 91. Admits a partition of unity.

Algebraic-topological properties Several properties to do with homotopy groups require selecting a base point. You can interpret each of these properties as being required to hold at every base point, at some base point, or just requiring that the space is path connected. 92. Simply connected: The fundamental group is trivial. 93. Weakly contractible: All homotopy groups are trivial. 94. Locally contractible. 95. Semilocally semisimply connected: Any point has a neighbourhood N such that the inclusion N ↪ X induces a trivial homomorphism on the fundamental group based at that point. This is usually called 'semilocally simply connected', but this clashes with my terminology. 96. Has an abelian fundamental group. 97. Has a finitely generated fundamental group. 98. Has an uncountable fundamental group.

Miscellaneous properties 99. Alexandrov-discrete: Any intersection of open subsets is open. 100. Door space: Any subset is open or closed or both. 101. Submaximal: Every subset is locally closed, i.e. the intersection of an open set and a closed set. 102. Contains a simple curve: Admits an injective continuous map from the closed interval. 103. Has a cut-point: Contains a non-isolated point x such that if you remove x from its connected component, it is no longer connected. 104. Has a generic point: Contains a dense singleton subset. 105. Homogeneous: For any two points x, y, there is a homeomorphism X -> X carrying x to y. 106. Weakly homogeneous: For any two points x, y and any neighbourhoods U, V of x, y respectively, there exist smaller neighbourhoods U' and V' that are pointedly homeomorphic w.r.t. the base points x, y. 107. Totally heterogeneous: X admits only one homeomorphism X -> X. 108. Uniformly based: Has a basis of homeomorphic subsets. 109. Self-based: Has a basis of subsets homeomorphic to itself. 110. Resolvable: Is the union of disjoint dense subsets. 111. Strongly discrete: Every non-isolated point is an accumulation point of a subset whose points can be simultaneously separated by open neighbourhoods. 112. L-space: Hereditarily Lindelöf and not hereditarily separable. 113. S-space: Hereditarily separable and not hereditarily Lindelöf. 114. Polyadic: Is the continuous image of a power of a one-point compactification of a discrete space.

[bubbloquacious]

Awesome :) you did the 'easier' version of the game where you don't also come up with a space that doesn't have the property in the case the property is independent. That's alright though, it's all about playing in the way you think is the most fun.

Note that ℝⁿ is actually reducible; it is the union of the closed upper half-space and the closed lower half-space, say. The union of proper closed subsets does not have to be a disjoint union, then irreducibility would be the same as connectedness (with non-emptiness). In fact, any metric space is Hausdorff, and the only Hausdorff irreducible space is the one point space, which is either not resolvable or it admits a partition of unity, so irreducibility is contradictory.

Moreover, the open unit ball is in fact homeomorphic with ℝⁿ, so it is a Euclidean space. That being said, it's not terribly hard to make a non-Euclidean space; ℝⁿ plus an isolated point works.

As a final note, you can just remove the #-15 from round 8, that's why the property tells you to reroll it :p being homeomorphic to a space with property P is the same as having property P (that's what makes P a topological property!).

The Topology Game

Here's an updated list of topological properties that I use to play @topoillogical's Topology Game. The rules are as follows. You roll a random integer between -N and N inclusive, where N is the number of properties below (currently N = 114). If the number is negative, you get the negation of the property whose number is the magnitude of the number. You come up with a topological space satisfying this property. This was round 1.

For all future rounds, you start by rolling another property. If the new property follows from the previous properties, prove this. If the new property contradicts the previous properties, prove this. If the new property is independent of all the previous, come up with two spaces satisfying all the previously rolled properties, where one space has the new property and one space has its negation.

You 'win' the game if you manage to solve round 10. You can never truly lose as you can always come back to a given game.

List of 114 properties:

A note on terminology: The space a property refers to is always denoted by X. A neighbourhood of a subset A of X is a subset N of X such that A is contained in the interior of N. Any subset of X is implicitly equipped with the subspace topology.

0. Add your favourite property here to get a slightly higher chance of rolling it.

Property modifiers: These are listed separately, but some specific incarnations (like 'locally connected') are also in the rest of the list because they are more common. If you roll a property modifier, roll again to get the property that is modified. If you roll another modifier, your modifiers stack, so you can get something like 'locally retracts onto a space that contains a compact space'. 1. Locally P: Every point has a neighbourhood basis of P subsets. 2. Somewhere locally P: Some point has a neighbourhood basis of P subsets. 3. Semilocally P: Every point has a P neighbourhood. 4. Somewhere semilocally P: Contains some P subset with non-empty interior. 5. Has a basis of P sets. 6. Hereditarily P: Every subset of X is P. 7. Weakly hereditarily P: Every closed subset of X is P. 8. Contains a P space: X contains some P subset. 9. Retracts onto a P space: X contains a P subset A as a retract, i.e. the inclusion A ↪ X has a continuous left inverse. 10. Is the image of a P space: There is a P space Y and a continuous surjection Y -> X. 11. Homotopy-equivalent to a P space. 12. Weakly homotopy-equivalent to a P space: There is a P space Y and a continuous map X -> Y that induces an isomorphism on all homotopy groups. 13. Is a projective limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical limit. 14. Is an inductive limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical colimit. 15. Is homeomorphic to a space Y: Here Y can be any topological space, for which you will need to find some other way to generate it. Reroll this property if you did not in this round first roll a modifier before.

Classes of spaces 16. Euclidean space. Included for completion, but probably not very interesting to roll. 17. Locally Euclidean. 18. Locally Euclidean with boundary. 19. Homeomorphic to a CW-complex. 20. Homotopy-equivalent to a CW-complex. 21. Homeomorphic to a linearly ordered set with the order topology. 22. Homeomorphic to a topological group. 23. Homeomorphic to a space of continuous maps with the compact-open topology. 24. Spectral: Is homeomorphic to the prime spectrum of a commutative ring with its Zariski topology. 25. Homeomorphic to a vector bundle of positive rank. 26. Homeomorphic to a non-trivial principal bundle.

Separation properties Two subsets A, B are said to be 'separated by P sets' if they have disjoint P neigbourhoods. A 'Urysohn function' for A and B is a function f from X to the real numbers such that A is contained in the preimage of 0 and B is contained in the preimage of 1. A 'perfect Urysohn function' for A and B is a Urysohn function such that A and B equal the preimages of 0 and 1 respectively. If you want you can replace some of the following properties with a weaker version that only holds between topologically distinguishable points, so that your property plus Kolmogorov becomes the property as written. 27. Indiscrete. Included for completion, but probably not very interesting to roll. 28. Point-distinguishing (Kolmogorov): No two points are contained in precisely the same open sets, i.e. any two points are topologically distinguishable. 29. Accessible (Fréchet): Any two points have open neighbourhoods not containing the other point. Equivalently, all singletons are closed. 30. Sober: Any irreducible closed set is the closure of a unique point. 31. Weakly Hausdorff: Any image of a compact Hausdorff space is closed. 32. All compact subsets are closed. 33. Hausdorff: Any two points are separated by open neighbourhoods. 34. Closed Hausdorff: Any two points are separated by closed neighbourhoods. 35. Urysohn Hausdorff: Any two points have a Urysohn function. 36. Perfectly Hausdorff: Any two points have a perfect Urysohn function. 37. Semiregular: The regular open sets, i.e. open sets equal to the interior of their closure, form a base of the topology. 38. Quasiregular: Any non-empty open set contains a non-empty regular open set. 39. Regular: Any point and any closed set not containing it are separated by open neighbourhoods. Not to be confused with being a regular open or closed set. 40. Urysohn regular: Any point and any closed set not containing it have a Urysohn function. 41. Perfectly regular: Any point and any closed set not containing it have a perfect Urysohn function. 42. Normal: Any two disjoint closed sets are separated by open neighbourhoods. Urysohn's lemma states exactly that this is equivalent to Urysohn normal. 43. Hereditarily normal. 44. Fully normal: Any open cover has an open star refinement. 45. Perfectly normal: Any two disjoint closed sets have a perfect Urysohn function. 46. Discrete. Included for completion, but probably not very interesting to roll.

Metrizability properties 47. Metrizable: Is homeomorphic to a metric space. 48. Completely metrizable: Is homeomorphic to a complete metric space. 49. Completely uniformizable: Is homeomorphic to a complete uniform space. Note that 'uniformizable' is equivalent to 'Urysohn regular'. 50. Developable: Admits a countable collection of open covers such that for any closed subset F and point x not in F, one of these covers satisfies that no set in the cover that contains x also intersects F.

Connectedness properties 51. Connected: Has exactly one connected component. 52. Locally connected. 53. Semilocally connected. 54. Path-connected: Any two points are connected by a path. 55. Simple path-connected: Any two distinct points are connected by a simple path, i.e. one that is injective. 56. Locally path-connected. 57. Contractible: Is homotopy-equivalent to a point. 58. Hyperconnected: No two non-empty open subsets are disjoint. 59. Ultraconnected: No two non-empty closed subsets are disjoint. 60. Irreducible: Not the union of two proper closed sets. 61. Dense-in-itself: There are no isolated points. 62. Scattered: Every non-empty subspace has at least one isolated point. 63. Zero-dimensional: The topology has a basis of clopen sets. 64. Totally disconnected: The only connected components are points.

Countability properties 65. Finite: Contains finitely many points. 66. Countable: Contains countably many points. 67. Continuum-sized: Admits a bijection with the real numbers. 68. Supercontinuum-sized: Larger than continuum-sized. 69. First countable: Every point has a countable neighbourhood basis. 70. Second countable: The topology has a countable basis. 71. Separable: Contains a countable dense subset. 72. Sequential: Any set closed under limits of sequences is closed. 73. Baire: If a countable union of closed sets has non-empty interior, then one of the closed sets had non-empty interior. 74. Meagre: Is a countable union of nowhere-dense subsets, i.e. subsets whose closures have empty interior. 75. Gδ space: Every closed set is a countable intersection of open sets. 76. Cosmic: Is the continuous image of a separable metric space.

Compactness properties 77. Compact: Any open cover has a finite subcover. 78. Semilocally compact. Note that this is usually called 'locally compact', but this clashes with my terminology. 79. Lindelöf: Any open cover has a countable subcover. 80. Sequentially compact: Any sequence has a convergent subsequence. 81. Limit point compact: Every infinite subset has a limit point. 82. Pseudocompact: Every real-valued continuous function on X is bounded. 83. σ-compact: Is a countable union of compact subsets. 84. Rim-compact: The topology has a basis of open sets with compact boundaries. 85. Core-compact: For every point x and every neighbourhood U of x there exists a smaller open neighbourhood whose closure in U is compact. 86. Paracompact: Any open cover has a locally finite refinement. 87. Metacompact: Any open cover has a refinement such that every point is contained in only finitely many sets of the refinement. 88. Orthocompact: Any open cover has a refinement such that for every point, the intersection of all open sets in the refinement containing that point is open. 89. Compactly generated: A subset U is open if and only if its intersection with any compact subset K is open in K. 90. Noetherian: Any downward chain of closed sets stabilizes. 91. Admits a partition of unity.

Algebraic-topological properties Several properties to do with homotopy groups require selecting a base point. You can interpret each of these properties as being required to hold at every base point, at some base point, or just requiring that the space is path connected. 92. Simply connected: The fundamental group is trivial. 93. Weakly contractible: All homotopy groups are trivial. 94. Locally contractible. 95. Semilocally semisimply connected: Any point has a neighbourhood N such that the inclusion N ↪ X induces a trivial homomorphism on the fundamental group based at that point. This is usually called 'semilocally simply connected', but this clashes with my terminology. 96. Has an abelian fundamental group. 97. Has a finitely generated fundamental group. 98. Has an uncountable fundamental group.

Miscellaneous properties 99. Alexandrov-discrete: Any intersection of open subsets is open. 100. Door space: Any subset is open or closed or both. 101. Submaximal: Every subset is locally closed, i.e. the intersection of an open set and a closed set. 102. Contains a simple curve: Admits an injective continuous map from the closed interval. 103. Has a cut-point: Contains a non-isolated point x such that if you remove x from its connected component, it is no longer connected. 104. Has a generic point: Contains a dense singleton subset. 105. Homogeneous: For any two points x, y, there is a homeomorphism X -> X carrying x to y. 106. Weakly homogeneous: For any two points x, y and any neighbourhoods U, V of x, y respectively, there exist smaller neighbourhoods U' and V' that are pointedly homeomorphic w.r.t. the base points x, y. 107. Totally heterogeneous: X admits only one homeomorphism X -> X. 108. Uniformly based: Has a basis of homeomorphic subsets. 109. Self-based: Has a basis of subsets homeomorphic to itself. 110. Resolvable: Is the union of disjoint dense subsets. 111. Strongly discrete: Every non-isolated point is an accumulation point of a subset whose points can be simultaneously separated by open neighbourhoods. 112. L-space: Hereditarily Lindelöf and not hereditarily separable. 113. S-space: Hereditarily separable and not hereditarily Lindelöf. 114. Polyadic: Is the continuous image of a power of a one-point compactification of a discrete space.

[bubbloquacious]

Yea, it contradicts your previous properties (in fact, it contradicts the empty set of properties), so you would go to the next round assuming its negation (or equivalently not assuming anything new at all).

The Topology Game

Here's an updated list of topological properties that I use to play @topoillogical's Topology Game. The rules are as follows. You roll a random integer between -N and N inclusive, where N is the number of properties below (currently N = 114). If the number is negative, you get the negation of the property whose number is the magnitude of the number. You come up with a topological space satisfying this property. This was round 1.

For all future rounds, you start by rolling another property. If the new property follows from the previous properties, prove this. If the new property contradicts the previous properties, prove this. If the new property is independent of all the previous, come up with two spaces satisfying all the previously rolled properties, where one space has the new property and one space has its negation.

You 'win' the game if you manage to solve round 10. You can never truly lose as you can always come back to a given game.

List of 114 properties:

A note on terminology: The space a property refers to is always denoted by X. A neighbourhood of a subset A of X is a subset N of X such that A is contained in the interior of N. Any subset of X is implicitly equipped with the subspace topology.

0. Add your favourite property here to get a slightly higher chance of rolling it.

Property modifiers: These are listed separately, but some specific incarnations (like 'locally connected') are also in the rest of the list because they are more common. If you roll a property modifier, roll again to get the property that is modified. If you roll another modifier, your modifiers stack, so you can get something like 'locally retracts onto a space that contains a compact space'. 1. Locally P: Every point has a neighbourhood basis of P subsets. 2. Somewhere locally P: Some point has a neighbourhood basis of P subsets. 3. Semilocally P: Every point has a P neighbourhood. 4. Somewhere semilocally P: Contains some P subset with non-empty interior. 5. Has a basis of P sets. 6. Hereditarily P: Every subset of X is P. 7. Weakly hereditarily P: Every closed subset of X is P. 8. Contains a P space: X contains some P subset. 9. Retracts onto a P space: X contains a P subset A as a retract, i.e. the inclusion A ↪ X has a continuous left inverse. 10. Is the image of a P space: There is a P space Y and a continuous surjection Y -> X. 11. Homotopy-equivalent to a P space. 12. Weakly homotopy-equivalent to a P space: There is a P space Y and a continuous map X -> Y that induces an isomorphism on all homotopy groups. 13. Is a projective limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical limit. 14. Is an inductive limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical colimit. 15. Is homeomorphic to a space Y: Here Y can be any topological space, for which you will need to find some other way to generate it. Reroll this property if you did not in this round first roll a modifier before.

Classes of spaces 16. Euclidean space. Included for completion, but probably not very interesting to roll. 17. Locally Euclidean. 18. Locally Euclidean with boundary. 19. Homeomorphic to a CW-complex. 20. Homotopy-equivalent to a CW-complex. 21. Homeomorphic to a linearly ordered set with the order topology. 22. Homeomorphic to a topological group. 23. Homeomorphic to a space of continuous maps with the compact-open topology. 24. Spectral: Is homeomorphic to the prime spectrum of a commutative ring with its Zariski topology. 25. Homeomorphic to a vector bundle of positive rank. 26. Homeomorphic to a non-trivial principal bundle.

Separation properties Two subsets A, B are said to be 'separated by P sets' if they have disjoint P neigbourhoods. A 'Urysohn function' for A and B is a function f from X to the real numbers such that A is contained in the preimage of 0 and B is contained in the preimage of 1. A 'perfect Urysohn function' for A and B is a Urysohn function such that A and B equal the preimages of 0 and 1 respectively. If you want you can replace some of the following properties with a weaker version that only holds between topologically distinguishable points, so that your property plus Kolmogorov becomes the property as written. 27. Indiscrete. Included for completion, but probably not very interesting to roll. 28. Point-distinguishing (Kolmogorov): No two points are contained in precisely the same open sets, i.e. any two points are topologically distinguishable. 29. Accessible (Fréchet): Any two points have open neighbourhoods not containing the other point. Equivalently, all singletons are closed. 30. Sober: Any irreducible closed set is the closure of a unique point. 31. Weakly Hausdorff: Any image of a compact Hausdorff space is closed. 32. All compact subsets are closed. 33. Hausdorff: Any two points are separated by open neighbourhoods. 34. Closed Hausdorff: Any two points are separated by closed neighbourhoods. 35. Urysohn Hausdorff: Any two points have a Urysohn function. 36. Perfectly Hausdorff: Any two points have a perfect Urysohn function. 37. Semiregular: The regular open sets, i.e. open sets equal to the interior of their closure, form a base of the topology. 38. Quasiregular: Any non-empty open set contains a non-empty regular open set. 39. Regular: Any point and any closed set not containing it are separated by open neighbourhoods. Not to be confused with being a regular open or closed set. 40. Urysohn regular: Any point and any closed set not containing it have a Urysohn function. 41. Perfectly regular: Any point and any closed set not containing it have a perfect Urysohn function. 42. Normal: Any two disjoint closed sets are separated by open neighbourhoods. Urysohn's lemma states exactly that this is equivalent to Urysohn normal. 43. Hereditarily normal. 44. Fully normal: Any open cover has an open star refinement. 45. Perfectly normal: Any two disjoint closed sets have a perfect Urysohn function. 46. Discrete. Included for completion, but probably not very interesting to roll.

Metrizability properties 47. Metrizable: Is homeomorphic to a metric space. 48. Completely metrizable: Is homeomorphic to a complete metric space. 49. Completely uniformizable: Is homeomorphic to a complete uniform space. Note that 'uniformizable' is equivalent to 'Urysohn regular'. 50. Developable: Admits a countable collection of open covers such that for any closed subset F and point x not in F, one of these covers satisfies that no set in the cover that contains x also intersects F.

Connectedness properties 51. Connected: Has exactly one connected component. 52. Locally connected. 53. Semilocally connected. 54. Path-connected: Any two points are connected by a path. 55. Simple path-connected: Any two distinct points are connected by a simple path, i.e. one that is injective. 56. Locally path-connected. 57. Contractible: Is homotopy-equivalent to a point. 58. Hyperconnected: No two non-empty open subsets are disjoint. 59. Ultraconnected: No two non-empty closed subsets are disjoint. 60. Irreducible: Not the union of two proper closed sets. 61. Dense-in-itself: There are no isolated points. 62. Scattered: Every non-empty subspace has at least one isolated point. 63. Zero-dimensional: The topology has a basis of clopen sets. 64. Totally disconnected: The only connected components are points.

Countability properties 65. Finite: Contains finitely many points. 66. Countable: Contains countably many points. 67. Continuum-sized: Admits a bijection with the real numbers. 68. Supercontinuum-sized: Larger than continuum-sized. 69. First countable: Every point has a countable neighbourhood basis. 70. Second countable: The topology has a countable basis. 71. Separable: Contains a countable dense subset. 72. Sequential: Any set closed under limits of sequences is closed. 73. Baire: If a countable union of closed sets has non-empty interior, then one of the closed sets had non-empty interior. 74. Meagre: Is a countable union of nowhere-dense subsets, i.e. subsets whose closures have empty interior. 75. Gδ space: Every closed set is a countable intersection of open sets. 76. Cosmic: Is the continuous image of a separable metric space.

Compactness properties 77. Compact: Any open cover has a finite subcover. 78. Semilocally compact. Note that this is usually called 'locally compact', but this clashes with my terminology. 79. Lindelöf: Any open cover has a countable subcover. 80. Sequentially compact: Any sequence has a convergent subsequence. 81. Limit point compact: Every infinite subset has a limit point. 82. Pseudocompact: Every real-valued continuous function on X is bounded. 83. σ-compact: Is a countable union of compact subsets. 84. Rim-compact: The topology has a basis of open sets with compact boundaries. 85. Core-compact: For every point x and every neighbourhood U of x there exists a smaller open neighbourhood whose closure in U is compact. 86. Paracompact: Any open cover has a locally finite refinement. 87. Metacompact: Any open cover has a refinement such that every point is contained in only finitely many sets of the refinement. 88. Orthocompact: Any open cover has a refinement such that for every point, the intersection of all open sets in the refinement containing that point is open. 89. Compactly generated: A subset U is open if and only if its intersection with any compact subset K is open in K. 90. Noetherian: Any downward chain of closed sets stabilizes. 91. Admits a partition of unity.

Algebraic-topological properties Several properties to do with homotopy groups require selecting a base point. You can interpret each of these properties as being required to hold at every base point, at some base point, or just requiring that the space is path connected. 92. Simply connected: The fundamental group is trivial. 93. Weakly contractible: All homotopy groups are trivial. 94. Locally contractible. 95. Semilocally semisimply connected: Any point has a neighbourhood N such that the inclusion N ↪ X induces a trivial homomorphism on the fundamental group based at that point. This is usually called 'semilocally simply connected', but this clashes with my terminology. 96. Has an abelian fundamental group. 97. Has a finitely generated fundamental group. 98. Has an uncountable fundamental group.

Miscellaneous properties 99. Alexandrov-discrete: Any intersection of open subsets is open. 100. Door space: Any subset is open or closed or both. 101. Submaximal: Every subset is locally closed, i.e. the intersection of an open set and a closed set. 102. Contains a simple curve: Admits an injective continuous map from the closed interval. 103. Has a cut-point: Contains a non-isolated point x such that if you remove x from its connected component, it is no longer connected. 104. Has a generic point: Contains a dense singleton subset. 105. Homogeneous: For any two points x, y, there is a homeomorphism X -> X carrying x to y. 106. Weakly homogeneous: For any two points x, y and any neighbourhoods U, V of x, y respectively, there exist smaller neighbourhoods U' and V' that are pointedly homeomorphic w.r.t. the base points x, y. 107. Totally heterogeneous: X admits only one homeomorphism X -> X. 108. Uniformly based: Has a basis of homeomorphic subsets. 109. Self-based: Has a basis of subsets homeomorphic to itself. 110. Resolvable: Is the union of disjoint dense subsets. 111. Strongly discrete: Every non-isolated point is an accumulation point of a subset whose points can be simultaneously separated by open neighbourhoods. 112. L-space: Hereditarily Lindelöf and not hereditarily separable. 113. S-space: Hereditarily separable and not hereditarily Lindelöf. 114. Polyadic: Is the continuous image of a power of a one-point compactification of a discrete space.

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20 examples of periodic solutions to the three-body problem