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I think this whole debate is a good springboard to talk about how mathematicians come up with rigorous definitions. They don't really come from a vacuum, especially our definition of a hole. So how did we come up with it and why is this definition better than any other?

Part of this discussion would be "why do we care that the number of holes are preserved under continuous deformation?" On a more formal note, just because we can describe holes are using topology, why does that mean they should be a homotopy invariant? There are at least a few topological properties which are invariant under homeomorphism but not homotopy equivalence. Compactness: ℝ is contractible so is homotopy equivalent to a point but ℝ is not compact whilst the one point space is. Orientability: removing a point from the torus and from the Klein bottle results in a space homotopy equivalent to a figure 8. So why should we expect that of holes? This is in no way to disagree with you op, I just think it's an interesting discussion more so than what the actual answer is.

A straw has one hole.

Strictly and topologically defined, there is only one hole that goes through the length of the straw.

Think of a CD. A CD only has one hole. Now, make it radially thinner. Stretch it out. Now all of a sudden, you have a straw. Does it still have one hole, or does it have two now? If it has two, at what point does it start having two? What would be a meaningful, rigorous definition to be able to say that the straw has two holes while the CD does not? Does the CD have two holes? What meaningful proof backs that up? We already have a topological definition of a hole. Everything points towards the CD and straw having one hole. Hell, a mug and donut, the cover art of topology, have the same topology as the CD and straw. And they both have one hole, so by extension, they must all have only one hole.

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Encountered today: a math professor wearing a shirt his grad students made for him, depicting him as a Moomin saying “I will literally kill myself if I don’t mention the orthogonal group”

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Topological Spaces 12: Important Examples

We have spent quite a while developing a lot of theory and now it is time to apply it to some very important spaces that show up time and time again in topology!

Here is the link to the previous post about quotient spaces. They show up a lot in what follows, as does a lot of our theory so I highly recommend being familiar with the previous posts before reading this one!

12.1: The n-Sphere, Sⁿ

Definition 12.1:

A 2-sphere:

Proposition 12.2:

Remark: The last part of the proof fails for n=0 since ℝ\{0} is not connected. And indeed, S⁰={-1,1} is not connected (in this topology).

12.2: The n-Torus, Tⁿ

Defintion 12.3:

A (2-)torus:

Proposition 12.4:

12.3: Identifaction Squares

A particularly useful diagram in topology is the identification diagram which shows [0,1]×[0,1] as a square with arrows to indicate how to glue sections of the boundary together. The number of arrows indicated which sides to glue together (no arrows means that section is not glued to anything) and the direction corresponds to which direction the sections should be glued together. This gluing can we formalised with the use of an equivalence relation.

Since [0,1]×[0,1] is compact and path connected, our resulting spaces will also be compact and path connected.

Example 12.5:

As the picture suggests, M is homeomorphic to a subset of ℝ³ meaning it inherets Hausdorffness and second countability. For sake of brievity, the proof is omitted.

Example 12.6:

Example 12.7:

It is possible to show that K is homeomorphic to a subspace of ℝ⁴ so it is Hausdorff and second countable. It is also possible to show that it cannot be homeomorphic to a subspace of ℝ³ but this requires more advanced theory.

Example 12.8:

Just like with K, it is possible to show that P is homeomorphic ℝ⁴ so it is Hausdorff and second countable, but is not homeomorphic to a subspace of ℝ³.

12.4: Real Projective Space

Definition 12.9:

ℝℙⁿ is path connected and compact by Prop 11.5 and Prop 12.2. It is possbile to show ℝℙⁿ is Hausdorff and second countable but this takes a little more thoery.

It is also possible to show that P≅ℝℙ².

The next and final post will be a summary of what we've seen as well as hints at what else there is to topology!

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Topological Spaces 13: Conclusions and Remaining Questions

We have almost come to the end of this series but first I'd like to give a little overview of what we've done and what more there is to topology!

I've put a few book recommendations at the end too!

Here is the link to the previous post.

In generalising the notion of continuity, we found a notion in which topological spaces are "the same" (more formally, it is possible to show that homeomorphisms define an equivalence relation on class of topological spaces). We have also found certain properties of spaces which remain invariant under homeomorphisms:

(path-)connectedness

compactness

Hausdorffness

Second Countability

These invariants allow us to tell when spaces aren't homeomorphic. Having these invariants is extremely helpful since the task of classifying topological spaces up to homeomorphism is an insurmountable one, even when restricting to "nice" spaces. These are not the only homeormophism invariants, just the common ones.

There are a couple of questions that we haven't really answered in these posts:

How do we know the spaces in post 12 are not all homeomorphic?

When is ℝⁿ≅ℝᵐ?

Although the spaces in post 12 look different, how do we show that rigorously? How do we know that S¹ isn't homeomorphic to S²? One tool is orientability. Loosely, a space is orientable if no matter how you traverse the space, your orientation is preserved. It turns out orientability is a homeomorphism invariant. Spheres and tori are orientable whereas the Möbius band, Klein bottle, and real projective spaces are non-orientable. Another invariant is something called the Euler characteristic which is a number associated to (some) topological spaces which helps us further distiguish between these spaces.

Another approach is to associate algebraic structures to topological spaces in such a way that homeomorphisms induce isomorphisms between those algebraic structures. This is the realm of algebraic topology and provides us with machinary sophisticated enough to prove a lot of things which are otherwise extremely hard to prove.

The second question may seem obvious. It must be only when n=m right? That is indeed the case, but how do we prove that? In the case where n=1, we can easily show that by removing a point from ℝ and ℝᵐ for m>1. ℝ\{0} is not connected but ℝᵐ\{0} is so they are not homeomorphic, so we can conclude that ℝ and ℝᵐ can't be homeomorphic. Technically we must show that if f:X->Y is a homeomorphism and A⊆X, then A is homeomorphic to f(A). However, doing this in general is not so easy. Sure, we could go to ℝ² and remove a line and say that ℝᵐ with a line removed is still connected but ℝ² isn't and then go to ℝ³ and repeat with a higher dimensional subspace. But this strategy of going case by case is not productive and would literally take forever. So is there a quicker way? Using something called homology groups, we find that ℝᵐ\{0} and ℝⁿ\{0} have different homology groups and therefore cannot be homeomorphic. Of course that is skipping over a tonne of theory.

Book recommendations:

Introduction to Metric and Topological Spaces by Wilson A. Sutherland. The structure of this series is heavily influenced by this book. There is an emphasis on building intuition for definitions and is great first reading of topology (in my opinion)

Topology by James Munkres. This book is full of a lot more than what we've covered here. Whilst I haven't read it cover to cover, I've found the sections I have read have been clear.

There's not anything else that I'd like to talk about in this post. If anyone has any questions about topology, don't be afraid to ask! The whole reason I started this post series is a mutual sent in an ask about topology.

[lipshits-continuous]

Topological Spaces 12: Important Examples

We have spent quite a while developing a lot of theory and now it is time to apply it to some very important spaces that show up time and time again in topology!

Here is the link to the previous post about quotient spaces. They show up a lot in what follows, as does a lot of our theory so I highly recommend being familiar with the previous posts before reading this one!

12.1: The n-Sphere, Sⁿ

Definition 12.1:

A 2-sphere:

Proposition 12.2:

Remark: The last part of the proof fails for n=0 since ℝ\{0} is not connected. And indeed, S⁰={-1,1} is not connected (in this topology).

12.2: The n-Torus, Tⁿ

Defintion 12.3:

A (2-)torus:

Proposition 12.4:

12.3: Identifaction Squares

A particularly useful diagram in topology is the identification diagram which shows [0,1]×[0,1] as a square with arrows to indicate how to glue sections of the boundary together. The number of arrows indicated which sides to glue together (no arrows means that section is not glued to anything) and the direction corresponds to which direction the sections should be glued together. This gluing can we formalised with the use of an equivalence relation.

Since [0,1]×[0,1] is compact and path connected, our resulting spaces will also be compact and path connected.

Example 12.5:

As the picture suggests, M is homeomorphic to a subset of ℝ³ meaning it inherets Hausdorffness and second countability. For sake of brievity, the proof is omitted.

Example 12.6:

Example 12.7:

It is possible to show that K is homeomorphic to a subspace of ℝ⁴ so it is Hausdorff and second countable. It is also possible to show that it cannot be homeomorphic to a subspace of ℝ³ but this requires more advanced theory.

Example 12.8:

Just like with K, it is possible to show that P is homeomorphic ℝ⁴ so it is Hausdorff and second countable, but is not homeomorphic to a subspace of ℝ³.

12.4: Real Projective Space

Definition 12.9:

ℝℙⁿ is path connected and compact by Prop 11.5 and Prop 12.2. It is possbile to show ℝℙⁿ is Hausdorff and second countable but this takes a little more thoery.

It is also possible to show that P≅ℝℙ².

The next and final post will be a summary of what we've seen as well as hints at what else there is to topology!

[lipshits-continuous]

Watched one video about crocheting the hyperbolic plane and now I have to physically hold myself back from burning money on yet another hobby (which I would suck ass at anyway because I have no dexterity or patience)

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The penultimate post in the intro to topology series is about 2/3 to 3/4 done and I will probably also write the last post tomorrow as that's more of a summary

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"Letters weren't supposed to be used in maths. Maths would be easier if it didn't have letters." Actually it would be harder.

I appreciate abstraction isn't easy for everyone but mathematics is better off for it.

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This has caused great debate among my math friends so I think it needs tumblr's input

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Topological Spaces 11: Quotient Spaces

Like product spaces, quotient spaces are another way of creating new topological spaces out of old ones. In this case, we are putting a topology on equivalence classes of elements in the original space. I will be assuming knowledge of equivalence relations and equivalence classes, so here is the wikipedia page about them. Unlike for product spaces, not all of the properties of the original space will necessarily be inhereted by the quotient space.

Quotient spaces play a very important role in topology as they can be used to formalise the notion of "gluing" spaces together.

We will not make any reference to anything from the previous two posts so it is possible to read this one without them.

Definition 11.1:

The topology we want to put on X/~ is one where U is open if it's preimage under the quotient map is open. The benefit of this is that we automatically get that the quotient map is continuous.

Proposition 11.2:

Before we give an example we shall prove a useful result!

Proposition 11.3:

Example 11.4:

Remark: We technically haven't shown that restricting a continuous function results in a continuous function, however the proof is relatively easy.

Proposition 11.5:

Remarks:

The converse to these are not true. Example 11.4 already provides a counterexample for compactness. For a counterexample to (path-)connectedness, let X be any space with two or more points which is not (path-)connected and let ~ be the equivalence relation x~y iff x∈X and y∈X. Then X/~ is the one pointed space, which is (path-)connected.

The same result does not hold for Hausdorffness and second countability. We don't have a result that says the continuous image of a Hausdorff/second countable space is Hausdorff/second countable so the same proof wouldn't apply. Whilst this doesn't prevent a different proof, it does key us into the fact that something else might be going on. The construction of counterexamples would require a bit more theory so are omitted but feel free to look them up for yourself!

We will see many more examples of quotient spaces in the next post, which is dedicated to a plethora of important examples of topological spaces!

[lipshits-continuous]

Topological Spaces 11: Quotient Spaces

Like product spaces, quotient spaces are another way of creating new topological spaces out of old ones. In this case, we are putting a topology on equivalence classes of elements in the original space. I will be assuming knowledge of equivalence relations and equivalence classes, so here is the wikipedia page about them. Unlike for product spaces, not all of the properties of the original space will necessarily be inhereted by the quotient space.

Quotient spaces play a very important role in topology as they can be used to formalise the notion of "gluing" spaces together.

We will not make any reference to anything from the previous two posts so it is possible to read this one without them.

Definition 11.1:

The topology we want to put on X/~ is one where U is open if it's preimage under the quotient map is open. The benefit of this is that we automatically get that the quotient map is continuous.

Proposition 11.2:

Before we give an example we shall prove a useful result!

Proposition 11.3:

Example 11.4:

Remark: We technically haven't shown that restricting a continuous function results in a continuous function, however the proof is relatively easy.

Proposition 11.5:

Remarks:

The converse to these are not true. Example 11.4 already provides a counterexample for compactness. For a counterexample to (path-)connectedness, let X be any space with two or more points which is not (path-)connected and let ~ be the equivalence relation x~y iff x∈X and y∈X. Then X/~ is the one pointed space, which is (path-)connected.

The same result does not hold for Hausdorffness and second countability. We don't have a result that says the continuous image of a Hausdorff/second countable space is Hausdorff/second countable so the same proof wouldn't apply. Whilst this doesn't prevent a different proof, it does key us into the fact that something else might be going on. The construction of counterexamples would require a bit more theory so are omitted but feel free to look them up for yourself!

We will see many more examples of quotient spaces in the next post, which is dedicated to a plethora of important examples of topological spaces!

[lipshits-continuous]

Introduction to Topology Master Post

Metric Spaces

Topological Spaces and Continuity

Closed Sets and Limit Points

Hausdorffness

Connectedness

Path Connectedness

Compactness

Bases and Second Countability

Product Spaces

The Heine-Borel Theorem

Quotient Spaces

Important Examples

Conclusions and Remaining Questions

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Topological Spaces 10: The Heine Borel Theorem

After out study of product spaces, we now have enough theory to prove the Heine-Borel Theorem is a relatively easy way. There are a couple of things we still need to prove before we can get to the proof.

Whilst we only use one result from the post about product spaces, I still recommend being familiar with it. You can read it here.

First we need two key results that we'll use in our proof.

Proposition 10.1:

Corollary 10.2:

Now we have everything we need to prove Heine-Borel!

Theorem 10.3:

In the next post we will be studying quotient spaces!

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“Suppose, god forbid, that you find yourself as a physicist…”

I love my pure lecturers

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