# 40

This is about the Math Textbook Topology from Miri's research guide. (You can find the pdf online for free.) I got this book about a year ago. It takes a rigorous bottom-up approach that requires almost no prior knowledge but a lot of time. It's long and there are many exercises. I've read most of the book and done most of the exercises in the parts I read. It taught me about topology, about proving theorems, and about being efficient with Latex.

Chapter 1: Set Theory and Logic

This is a general introduction to highest mathematics and has nothing to do with topology. It introduces fundamental concepts such as logical implications, sets, tuples, relations, and functions. I've worked through this perhaps more thoroughly than I needed to, but I got some real value out of it: the book makes some things explicit that are often brushed over, such as when and why one is allowed to use proofs by induction, or what hides behind the supremum operation on an ordered set, and when one is allowed to use it.

The most interesting part about this was the construction of the usual number sets. Rather than beginning by defining (for example through as is done in ZFC), it starts by asserting the existence of a set called the real numbers, and of two operators and an order relation on which fulfil a list of eight axioms. From there, the sets and are constructed out of .

Two of the axioms on state that has the least-upper-bound property (which is precisely what is needed for the supremum) and that, given in , there is an element such that .

This approach is quite different from the ZFC construction: now is taken to be the most fundamental set rather than something one needs to be constructed through a sequence of complicated steps. This intuition is compatible with the rest of the book: as a topological space, is a more standard example than . The approach also requires less work.

Chapter 2: Topological Spaces and Continuous Functions

A topological space is a pair where is any set and a set of subsets of , that is, , or equivalently . One can think of the topology as a bunch of bubbles covering the elements of . A topology must meet the following three axioms:

(1):

(2):

(3):

That is, the topology is closed under arbitrary unions (3) ( is any index set), but only finite intersections (2).

A subset is called open if and only if , and it is called closed if and only if (the is set-difference). Open is not the opposite of closed; a set can be open or closed or neither of both (like and ). A topology just is then just the collection of all open sets of . Before reading this book, insofar as I knew what open sets were at all, I used to think of them as sets where every point has a small area around it that is also in the set (such as ). But the topological definition also allows to be plus all sets that contain a fixed point , for example. As far as I know, this topology is not seriously "used" for anything, but it does meet all three axioms. Other strange examples exist.

The book mentions that it took a while to reach a consensus on what exactly a topological space should be, and this appears to be the most useful generalization of concepts from analysis. Here is a theorem which I find gives it a bit of intuition.

Theorem. Let be a topological space, and let . Then,

This reads "A set is open if and only if for each of its points, there exists an open set around that point which is contained in ". This might be closer to what one thinks being open means.

Proof. : given , one has . for each , let be an open set such that . Then , so is open.

The second step is using the fact that arbitrary unions of open sets are open. I remember feeling intuitively that if a bubble is put around every element in the set, then the union of all of these bubbles must be more than just the set itself. But if the bubbles are all contained in the set, then it's easy to prove that the union is precisely the set itself.

This theorem has frequent use: most of the time one wants to show that a set is open, one does it by picking an arbitrary element in the set and fitting an open set around it while staying within .

The standard topology on consists of all the sets that are unions of open intervals with in . Note that the intersection of all open sets containing a point is just the one-point set (this follows from the second axiom of that I listed). But the intersection of arbitrarily many points does not need to be open; only finite intersections need be open And indeed, is not open in (but it is closed).

One way to define a topology on a space is to define a metric that meets a bunch of properties and is supposed to be a coherent measure of distance between any two points of . The topology induced by consists of all sets for which there is an such that . Then is called a metric space, and given a function between two metric spaces, one can define continuity with the - definition from analysis. But the topological definition of continuity is more general than that. Given a function between two topological spaces, is defined to be continuous if and only if for every set open in , the set is open in . This is equivalent to the - definition in cases where is a metric with the topology induced by . It's more general because every metric induces a topology, but not every topology has a metric inducing it.

In most fields of mathematics, functions are a central focus (why is this?). Most (all?) fields take a special interest in some particular class of functions, which are usually just a tiny area in the space of all functions (take a continuous function from to and change any image point by any amount, and it's no longer continuous). In algebra, one wants to have functions that preserve structure; in the of a function between two abstract groups, one wants that . A function fulfilling this is then called a homomorphism. The analogous concept in topology and analysis is a continuous function: it doesn't preserve structure (there need not be any "structure" analogous to that induced by the operator on a topological space), but it preserves topology, which for metric spaces means that arbitrarily small changes of lead to arbitrarily small spaces of , and in the more general case of topology, that for every open set around there must be an open set around such that (this is equivalent to the requirement that be open, by the theorem I proved above). The analogous concept to an isomorphism between groups, which is a bijective homomorphism, is then a homeomorphism, which is a bijective continuous function such that is also continuous. If there exists a homeomorphism between two topological spaces , and , then they are called homeomorphic. In that case, they are said to be "topologically identical", since all properties which are formulated in terms of their topologies (such as the existence of continuous functions that do certain things) are equivalent for both. There are many such properties that are of interest.

This has always been somewhat unintuitive to me. Whether two spaces are homeomorphic depends on seemingly strange things; for example, the open interval (the term open has a different meaning for intervals than for sets in a topological space, but if is given the standard topology, they coincide) which might intuitively seem "small" is homeomorphic to the entire set of real numbers . Concepts like length are not topological; they can change under a homeomorphism. But fine, that's similar to familiar properties of infinity: the sets and don't feel like they're the same size, but they're bijective. Same for and . However, the half-open interval [0,1), it is no longer homeomorphic to . A single point has changed things, which is new: the spaces and are still bijective (even though writing down an explicit bijective function is tricky).

In the real world, if we go down to the smallest building blocks of the universe, then their impact also becomes arbitrarily small, I believe. This makes it seem implausible that a formal system where single points have so much importance is useful. But obviously, this intuition is wrong. For example, fixed point theorems seem to be of some importance in AI alignment, and those are fundamentally topological problems. The disc (that is, the space ) is often denoted , and one can prove that any continuous function has a fixed point, that is, there exists a point such that . This is a result that falls out of deeper studies of topology (though there are many different ways to prove it), and it also generalizes to the -dimensional ball . Once again, it is no longer true when one takes a point out of (if one takes out the center, for example, then the function rotating everything around the center is a continuous map without a fixed point). It is kind of amazing that the open problem Scott Garrabrant posted here requires (almost) no further tools to be formally stated than what is covered by the first two chapters of this book!

Perhaps the fundamental reason why my intuition is wrong is that we aren't trying to study nature and its messiness, but we are trying to figure out how to design systems, where we can achieve a very high degree of precision?

Chapter 3: Connectedness and Compactness

A space is connected if it can't be separated into two open sets. It is compact if every collection of open sets that covers it has a finite sub-collection that also covers it.

If is a topological space and , a limit point of is a point such that for every with intersects (the point is then 'arbitrarily close' to the rest of ; it might or might not lie in ). The set of plus all of its limit points is denoted ; it is the same as the intersection of all closed sets that contain . If is compact, then every infinite set in has a limit point. In a metric space, the reverse is also true. Closed subsets of compact spaces are compact.

Connectedness and compactness are "topological properties", they are preserved under a homeomorphism. This fact can then be used to prove that and are not homeomorphic: the set is compact but isn't (the sequence has no limit point). Similarly, while they are both connected, you can take the point or out of and it is still connected, but taking out any point of leaves an unconnected space (and if there were a homeomorphism between them, then would also be a homeomorphism). Connectedness can also be used to show that and aren't homeomorphic.

Theorem. Let be a compact metric space. Let be a map such that for all in . Then has a unique fixed point.

Proof. This was difficult for me at the time. (Some adjustments made to readability, but not to the chain of arguments.) Is there a qualitatively shorter way? I don't know.

This is less powerful than the fixed point theorems for because it demands that has this property, but the upshot is that it works for every compact metric space.

Meta-insight for proving theorems: always have pen and paper, always make little drawings. It's low effort and almost always helps.

Chapter 4: Countability and Separation Axioms

Since the definition of a topological space is so general, there are a bunch of properties that feel useful but aren't always met. So mathematicians have defined them and given them names. Now, if one can prove that they are met, a number of useful results are immediate.

The Hausdorff property states that for any two different points and in a topological space , there exist open sets such that and and . Regularity demands the same (two disjoint open sets) for a point and a closed set ; normality for two closed sets.

Theorem. Let be a topological space. If is compact and Hausdorff, then is normal.

Proof. (Skit.) We first show that is regular. Let be closed and let . For each , choose disjoint open sets and such that and . The collection covers . Choose a finite subcollection that also covers (closed subsets of compact spaces are compact). Then is an open set around and a disjoint open set around . Thus, is regular. To prove normality, given closed sets , repeat the argument with open sets around each and disjoint open sets around (use regularity). //

Theorem. Let and be topological spaces, let be Hausdorff. Let be continuous. Then the graph of defined by is closed in the product space .

Proof. (Skit.) I haven't defined the topology on a product space here, though.

I've done this proof twice, once when I worked through Munkres' book, and once as an exercise in the lecture on topology I've taken the past semester. My second proof (the one above) is much shorter and also simpler. Does that mean I improved?

There is another theorem which states that a topological space is Hausdorff if and only if the diagonal is closed in . In the lecture, this was given on the same sheet as the exercise above. And indeed, using the result above, one of the implications becomes a triviality: the identity map is continuous, so is closed, and . In the spirit of the lecture, it was considered stupid to prove this using primitive arguments. It's far simpler with the above theorem! But in Munkres' book, it was an exercise in chapter 2, before functions were even introduced. And it was really difficult for me. Did the book waste my time?

I don't think so. The lecture tried to get away from primitive arguments as quickly as possible. Only use them if it is absolutely necessary, and optimize the structure of lecture and exercises for the ability to do everything as elegantly as possible. But why would that teach the right skillset? This has been on my mind a lot, and I think Munkres has the better idea. There is certainly a spectrum here, but optimizing for elegance only seems wrong.

Chapter 5: The Tychonoff Theorem

The Tychonoff theorem states that an arbitrary product of compact spaces is compact. The product topology is not the naively most obvious way to define a topology on a product space, however, and the result does not hold for product spaces in the box topology (although there are different & simpler reasons to prefer the product topology). The proof of this general result is far harder than the proof that finite products are compact.

The proof requires Zorn's Lemma, which is equivalent to the Axiom of Choice, which is the last axiom of ZFC (the "C" stands for "choice"). An alternative proof uses the Well-Ordering theorem, which is also equivalent to the Axiom of Choice.

Chapter 6: Metrization Theorems and Paracompactness

The metric topology is very well-behaved and understood. If one could prove about a topological space that there is a metric on such that induces (in short, if is metrizable), then one immediately gains a long list of useful properties that are met by . This is why theorems that find conditions on a space which imply metrizability are of interest. The first such result proves that regularity (separation axiom) and having a countable basis (countability axiom) together imply metrizability. A stronger result weakens the requirement of having a countable basis and proves logical equivalence of metrizability and regularity & having a basis that is countably locally finite.

The concept of local finiteness sounds odd but turns out to be useful. A collection of subsets of is locally finite if for every point there is an open set around which intersects only finitely many of them. The collection of all intervals is locally finite in but is not finite. There are also local versions of the properties compactness, metrizability, connectedness, and path-connectedness. In the latter two cases, neither of the two versions (normal and local) imply the other.

Chapter 7: Complete Metric Spaces and Function Spaces

This was the most difficult chapter for me. I find it hard to deal with sets of functions – a function can be thought of as a point in the infinite-dimensional space , but how does one visualize an (open or otherwise) set of such points? It is made more complicated still by the fact that there are as many as four different topologies introduced on function spaces. The "normal" one, that is, the topology one gets from simply imposing the product topology on the space corresponds to a sort of "point-wise" study of functions. In particular, a sequence of functions converges to a function in the product topology (convergence like continuity is a purely topological property) if and only if it converges point-wise (as defined in analysis). Similarly, it converges in the uniform topology if and only if it converges uniformly (as defined in analysis). Then there is the topology of compact convergence and the compact-open topology.

A metric space is called complete if every Cauchy sequence (= a sequence of points whose pairwise distances become and remain arbitrarily small) converges. The diameter of a set in a metric space is the supremum of pairwise distances in the set.

Theorem. A metric space is complete if and only if every sequence of closed nonempty sets such that has a nonempty intersection.

Proof. Skits. This is one of those rudimentary proofs that I think are good practice. The drawings are both for the second direction of the proof.

Chapter 8: Baire Spaces and Dimension Theory

I've only started this and done a few exercises. The definition of a Baire space is very unnatural feeling and I don't yet have any intuition of why it is useful.

Chapter 9: The Fundamental Group

This probably takes the cake as the hardest chapter, but I had an easier time with it than with chapter 7, because I found it truly fascinating – unlike ch7, which felt like more of a grind.

A path on a topological space is a continuous map . The points and are called the endpoints of , and is said to go from to . If , then is said to be a loop based at . The space is path-connected if there exists a path from to for any .

There is an equivalence relation on the set of all loops based at a fixed point , where iff there is a path homotopy between them, which can be thought of as a continuous deforming of into such that the base point remains fixed. On a convex vector space, any two paths are path homotopic, since one can just connect them pointwise by a straight line. But on the circle , the path that goes around the circle once is not path homotopic to the path that just sits at a single point. The base point has to remain fixed, so there is simply no way to undo the one circulation. One can think about a rubber band wrapped once around a disc; at any point, the band can be pulled apart to make it longer, but without undoing the base point or leaving the circle, it can't be reduced to a path that does zero circulations (or more than one).

Two paths and where (such as two loops with the same base point) can be connected by simply going along first and then . The resulting path is denoted . This operation can be proven to be well-defined on equivalence classes of paths under homotopy equivalence. Furthermore, for any path there exists a reverse path , and , where is a constant path. And with that, the set of all equivalence classes of loops based at with operation forms a group! It is called -t-h-e- a fundamental group of the space . Not 'the' because if is not path connected, then fundamental groups at different base points may be different.

My favorite thing about this is that every continuous function with defines a function via , and the function is a homomorphism between the two groups and . That means if preserves the topology, then preserves structure! And to make the analogy perfect: if is a homeomorphism, then is an isomorphism! And this is not only beautiful, but it also proves that the fundamental group is a topological invariant. Homeomorphic spaces have isomorphic fundamental groups, and the contrapositive statement is that if two spaces do not have isomorphic fundamental groups, then they are not homeomorphic. So the fundamental group is a way to prove that two spaces are 'topologically different'. It is more general than arguments based on the handful of topological properties studied previously, but not strictly more general; the spaces and both have the trivial fundamental group.

There's more. The fundamental group of the circle can be used to prove the fundamental theorem of algebra, which is pretty surprising and a strong knockdown to my concerns that theorems which care about single points can't be useful. It can also be used to prove that for any two bounded polygonal regions in , there is a single cut that divides both exactly in half.

The fundamental group of is isomorphic to the infinite cyclic group . The homotopy classes are exactly determined by how often each path goes around the circle (and it can go around it in two ways, hence the negative numbers). The fundamental group of the sphere is not homeomorphic to , but to the trivial group . There are also spaces that have non-trivial finite groups.

Chapter 10: Separation Theorems in the Plane

The last quarter of the book consists of much shorter chapters. I've only started this one, it (among other things) about how continuous maps always divide the plane into two regions, one bounded and the other unbounded.

Chapter 11: The Seifert-van Kampen Theorem

Chapter 12: Classification of Covering Spaces

Chapter 13: Classification of Surfaces

This was done extensively in the lecture, though in such a way that I felt like we didn't truly prove anything. This is a feeling I've never had reading this book!

Conclusion: This book is great. It's well structured, everything makes sense, everything is built neatly on top of each other, and the number of exercises leaves nothing to be desired. In general, I've had thoroughly positive experiences with Miri's guide; I've so far studied with four of the textbooks linked there, and all of them have been great (and the non-textbooks, too!). I don't do well learning out of source material that frustrate me (which happens a lot), so having a collection of high quality textbooks across a wide variety of topics has been extremely helpful. I'm planning to work through as much material as I can while I'm finishing my master's degree.

The worst thing I can say about this book is that it doesn't seem quite as impressive as Linear Algebra Done Right and Computability and Logic. In case of these two books (particularly the former), I've just been blown away by how much better and easier they are than my previous introductions to these topics. Nothing in this book gave me that impression, but as I said, it is still extremely solid. And it should be said that it covers a much larger and more difficult subject.