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 N (for example through n={0,...,n−1} as is done in ZFC), it starts by asserting the existence of a set R called the real numbers, and of two operators +,⋅:R2→R and an order relation < on R which fulfil a list of eight axioms. From there, the sets Z and Q are constructed out of R.

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

This approach is quite different from the ZFC construction: now R 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, R is a more standard example than N. The approach also requires less work.

Chapter 2: Topological Spaces and Continuous Functions

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

(1): ∅,X∈T

(2): ∀O1,...,On∈T:(O1∩...∩On)∈T

(3): ∀{Oj}j∈J⊂T:(⋃j∈JOj)∈T

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

A subset O⊂X is called open if and only if O∈T, and it is called closed if and only if (X−O)⊂T (the − is set-difference). Open is not the opposite of closed; a set can be open or closed or neither of both (like ∅ and X). A topology just is then just the collection of all open sets of X. 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 {(x,y)∈R:x2+y2<1}). But the topological definition also allows T to be ∅ plus all sets that contain a fixed point x0∈X, 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 (X,T) be a topological space, and let O⊂X. Then,

O∈T⟺∀x∈O∃U∈T:x∈U⊂O

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

Proof. ‘‘⟹": given x∈O, one has x∈O⊂O. ‘‘⟸": for each x∈X, let Ux be an open set such that x∈Ux⊂O. Then O=⋃x∈XUx, so O 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 A is open, one does it by picking an arbitrary element in the set and fitting an open set around it while staying within A.

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

One way to define a topology on a space X is to define a metric d:X2→R that meets a bunch of properties and is supposed to be a coherent measure of distance between any two points of X. The topology induced by d consists of all sets O⊂X for which there is an ϵ∈R such that Bd(x,ϵ):={p∈X:d(x,p)<ϵ}⊂O. Then (X,d) is called a metric space, and given a function between two metric spaces, one can define continuity with the ϵ-δ definition from analysis. But the topological definition of continuity is more general than that. Given a function f:X→Y between two topological spaces, f is defined to be continuous if and only if for every set O open in Y, the set f−1(O) is open in X. This is equivalent to the ϵ-δ definition in cases where X is a metric with the topology induced by d. 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 R to R 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 f(x∗y)=f(x)∗f(y). 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 x lead to arbitrarily small spaces of f(x), and in the more general case of topology, that for every open set U around f(x) there must be an open set O around x such that f(O)⊂U (this is equivalent to the requirement that f−1(U) 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 f such that f−1 is also continuous. If there exists a homeomorphism f:X→Y between two topological spaces X, and Y, 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 (0,1) (the term open has a different meaning for intervals than for sets in a topological space, but if R is given the standard topology, they coincide) which might intuitively seem "small" is homeomorphic to the entire set of real numbers R. Concepts like length are not topological; they can change under a homeomorphism. But fine, that's similar to familiar properties of infinity: the sets N and Q don't feel like they're the same size, but they're bijective. Same for (0,1) and R. However, the half-open interval [0,1), it is no longer homeomorphic to R. A single point has changed things, which is new: the spaces [0,1) and R 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 theoremsseem to be of some importance in AI alignment, and those are fundamentally topological problems. The disc (that is, the space {(x,y)∈R2:x2+y2≤1}) is often denoted D2, and one can prove that any continuous function f:D2→D2 has a fixed point, that is, there exists a point x∈D2 such that f(x)=x. 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 n-dimensional ball Dn. Once again, it is no longer true when one takes a point out of Dn (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 (X,T) is a topological space and A⊂X, a limit point of A is a point such that for every O⊂X with x∈O intersects A (the point x is then 'arbitrarily close' to the rest of A; it might or might not lie in A). The set of A plus all of its limit points is denoted ¯¯¯¯A; it is the same as the intersection of all closed sets that contain A. If X is compact, then every infinite set in X 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 [0,1] and (0,1) are not homeomorphic: the set [0,1] is compact but (0,1) isn't (the sequence (1n)n∈N+ has no limit point). Similarly, while they are both connected, you can take the point 0 or 1 out of [0,1] and it is still connected, but taking out any point of (0,1) leaves an unconnected space (and if there were a homeomorphism f between them, then f:[0,1]−{p}→(0,1)−{f(p)} would also be a homeomorphism). Connectedness can also be used to show that [0,1) and (0,1) aren't homeomorphic.

Theorem. Let (X,d) be a compact metric space. Let f:X→X be a map such that d(f(x),f(y))<d(x,y) for all x≠y in X. Then f 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 Dn because it demands that f 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 x and y in a topological space (X,T), there exist open sets O,U∈T such that x∈O and y∈U and O∩U=∅. Regularity demands the same (two disjoint open sets) for a point x and a closed set C; normality for two closed sets.

Theorem. Let (X,T) be a topological space. If X is compact and Hausdorff, then X is normal.

Proof. (Skit.) We first show that X is regular. Let C⊂X be closed and let x∈X−C. For each y∈C, choose disjoint open sets Uy and Oy such that x∈Uy and y∈Oy. The collection {Oy}y∈C covers C. Choose a finite subcollection O1,...,On that also covers C (closed subsets of compact spaces are compact). Then U1∩...∩Un is an open set around x and O1∪...∪On a disjoint open set around C. Thus, X is regular. To prove normality, given closed sets C,D⊂X, repeat the argument with open sets Ui around each x∈C and disjoint open sets Oi around D (use regularity). //

Theorem.Let X and Y be topological spaces, let Y be Hausdorff. Let f:X→Y be continuous. Then the graph of f defined by Γf={(x,f(x)):x∈X} is closed in the product space (X×Y).

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 X is Hausdorff if and only if the diagonalΔX={(x,x):x∈X)} is closed in X×X. 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 idX:X→X is continuous, so ΓidX is closed, and ΓidX=ΔX. 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 (X,T) that there is a metric d on X such that d induces T (in short, if (X,T) is metrizable), then one immediately gains a long list of useful properties that are met by X. 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 X is locally finite if for every point x∈X there is an open set around X which intersects only finitely many of them. The collection of all intervals (n,n+1) is locally finite in R 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 f:R→R can be thought of as a point in the infinite-dimensional space RR, 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 RR corresponds to a sort of "point-wise" study of functions. In particular, a sequence (fn)n∈N of functions converges to a function f 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 (X,d) is complete if and only if every sequence of closed nonempty sets A0⊃A1⊃... such that diam(Ai)→0 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 (X,T) is a continuous map p:[0,1]→X. The points p(0) and p(1) are called the endpoints of p, and p is said to go from p(0) to p(1). If p(0)=a=p(1), then p is said to be a loop based at a. The space is path-connected if there exists a path from x to y for any x,y∈X.

There is an equivalence relation on the set of all loops based at a fixed point b∈X, where p∼q iff there is a path homotopy between them, which can be thought of as a continuous deforming of p into q 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 S1, the path p:s↦e2πis that goes around the circle once is not path homotopic to the path q(s)≡(1,0) 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 p and q where p(1)=q(0) (such as two loops with the same base point) can be connected by simply going along p first and then q. The resulting path is denoted p∗q. This operation can be proven to be well-defined on equivalence classes [p] of paths under homotopy equivalence. Furthermore, for any path p there exists a reverse path ¯¯¯p, and [p∗¯¯¯p]=[c], where c is a constant path. And with that, the set π1(X,b) of all equivalence classes of loops based at b with operation ∗ forms a group! It is called -t-h-e- a fundamental group of the space X. Not 'the' because if X is not path connected, then fundamental groups at different base points may be different.

My favorite thing about this is that every continuous function f:X→Y with f(b0)=b1 defines a function f∗:π1(X,b0)→π1(Y,b1) via f∗([p])=[f∘p], and the function f∗ is a homomorphism between the two groups π1(X,b0) and π1(Y,b1). That means if f preserves the topology, then f∗ preserves structure! And to make the analogy perfect: if f is a homeomorphism, then f∗ 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 [0,1) and [0,1] 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 R2, there is a single cut that divides both exactly in half.

The fundamental group of S1 is isomorphic to the infinite cyclic group (Z,+). 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 S2 is not homeomorphic to (Z2,+), but to the trivial group ({1},∗). 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 f:S1→R2 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 bookis 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.

This is about the Math Textbook

Topologyfrom 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 LogicThis 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 N (for example through n={0,...,n−1} as is done in ZFC), it starts by asserting the existence of a set R called the real numbers, and of two operators +,⋅:R2→R and an order relation < on R which fulfil a list of eight axioms. From there, the sets Z and Q are constructed out of R.

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

This approach is quite different from the ZFC construction: now R 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, R is a more standard example than N. The approach also requires less work.

Chapter 2: Topological Spaces and Continuous FunctionsA

topological spaceis a pair (X,T) where X is any set and T a set of subsets of X, that is, T∈P(P(X)), or equivalently T⊂P(X). One can think of the topology as a bunch of bubbles covering the elements of X. A topology must meet the following three axioms:(1): ∅,X∈T

(2): ∀O1,...,On∈T:(O1∩...∩On)∈T

(3): ∀{Oj}j∈J⊂T:(⋃j∈JOj)∈T

That is, the topology is closed under

arbitraryunions (3) (J is any index set), but onlyfiniteintersections (2).A subset O⊂X is called

openif and only if O∈T, and it is calledclosedif and only if (X−O)⊂T (the − is set-difference). Open is not the opposite of closed; a set can be open or closed or neither of both (like ∅ and X). A topology just is then just the collection of all open sets of X. 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 {(x,y)∈R:x2+y2<1}). But the topological definition also allows T to be ∅ plus all sets that contain a fixed point x0∈X, 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 (X,T) be a topological space, and let O⊂X. Then,O∈T⟺∀x∈O∃U∈T:x∈U⊂O

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

Proof.‘‘⟹": given x∈O, one has x∈O⊂O. ‘‘⟸": for each x∈X, let Ux be an open set such that x∈Ux⊂O. Then O=⋃x∈XUx, so O 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 A is open, one does it by picking an arbitrary element in the set and fitting an open set around it while staying within A.

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

One way to define a topology on a space X is to define a metric d:X2→R that meets a bunch of properties and is supposed to be a coherent measure of distance between any two points of X. The topology induced by d consists of all sets O⊂X for which there is an ϵ∈R such that Bd(x,ϵ):={p∈X:d(x,p)<ϵ}⊂O. Then (X,d) is called a

metric space,and given a function between two metric spaces, one can define continuity with the ϵ-δ definition from analysis. But the topological definition of continuity is more general than that. Given a function f:X→Y between two topological spaces, f is defined to be continuous if and only if for every set O open in Y, the set f−1(O) is open in X. This is equivalent to the ϵ-δ definition in cases where X is a metric with the topology induced by d. 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 R to R 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 f(x∗y)=f(x)∗f(y). A function fulfilling this is then called ahomomorphism. The analogous concept in topology and analysis is acontinuous 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 preservestopology, which for metric spaces means that arbitrarily small changes of x lead to arbitrarily small spaces of f(x), and in the more general case of topology, that for every open set U around f(x) there must be an open set O around x such that f(O)⊂U (this is equivalent to the requirement that f−1(U) be open, by the theorem I proved above). The analogous concept to anisomorphismbetween groups, which is a bijective homomorphism, is then ahomeomorphism, which is a bijective continuous function f such that f−1 is also continuous. If there exists a homeomorphism f:X→Y between two topological spaces X, and Y, then they are calledhomeomorphic. 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 (0,1) (the term open has a different meaning for intervals than for sets in a topological space, but if R is given the standard topology, they coincide) which might intuitively seem "small" is homeomorphic to the entire set of real numbers R. Concepts like length are not topological; they can change under a homeomorphism. But fine, that's similar to familiar properties of infinity: the sets N and Q don't feel like they're the same size, but they're bijective. Same for (0,1) and R. However, the half-open interval [0,1), it is no longer homeomorphic to R. A single point has changed things, which is new: the spaces [0,1) and R 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 theoremsseem to be of some importance in AI alignment, and those are fundamentally topological problems. The

disc(that is, the space {(x,y)∈R2:x2+y2≤1}) is often denoted D2, and one can prove that any continuous function f:D2→D2 has a fixed point, that is, there exists a point x∈D2 such that f(x)=x. 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 n-dimensional ball Dn. Once again, it is no longer true when one takes a point out of Dn (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

designsystems, where we can achieve a very high degree of precision?Chapter 3: Connectedness and CompactnessA space is

connectedif it can't be separated into two open sets. It iscompactif every collection of open sets that covers it has a finite sub-collection that also covers it.If (X,T) is a topological space and A⊂X, a

limit pointof A is a point such that for every O⊂X with x∈O intersects A (the point x is then 'arbitrarily close' to the rest of A; it might or might not lie in A). The set of A plus all of its limit points is denoted ¯¯¯¯A; it is the same as the intersection of all closed sets that contain A. If X is compact, then every infinite set in X 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 [0,1] and (0,1) are not homeomorphic: the set [0,1] is compact but (0,1) isn't (the sequence (1n)n∈N+ has no limit point). Similarly, while they are both connected, you can take the point 0 or 1 out of [0,1] and it is still connected, but taking out any point of (0,1) leaves an unconnected space (and if there were a homeomorphism f between them, then f:[0,1]−{p}→(0,1)−{f(p)} would also be a homeomorphism). Connectedness can also be used to show that [0,1) and (0,1) aren't homeomorphic.

Theorem.Let (X,d) be a compact metric space. Let f:X→X be a map such that d(f(x),f(y))<d(x,y) for all x≠y in X. Then f 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 Dn because it demands that f 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 AxiomsSince 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

Hausdorffproperty states that for any two different points x and y in a topological space (X,T), there exist open sets O,U∈T such that x∈O and y∈U and O∩U=∅.Regularitydemands the same (two disjoint open sets) for a point x and a closed set C;normalityfor two closed sets.Theorem. Let (X,T) be a topological space. If X is compact and Hausdorff, then X is normal.Proof. (Skit.)We first show that X is regular. Let C⊂X be closed and let x∈X−C. For each y∈C, choose disjoint open sets Uy and Oy such that x∈Uy and y∈Oy. The collection {Oy}y∈C covers C. Choose a finite subcollection O1,...,On that also covers C (closed subsets of compact spaces are compact). Then U1∩...∩Un is an open set around x and O1∪...∪On a disjoint open set around C. Thus, X is regular. To prove normality, given closed sets C,D⊂X, repeat the argument with open sets Ui around each x∈C and disjoint open sets Oi around D (use regularity). //Theorem.Let X and Y be topological spaces, let Y be Hausdorff. Let f:X→Y be continuous. Then the graph of f defined by Γf={(x,f(x)):x∈X} is closed in the product space (X×Y).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 X is Hausdorff if and only if the

diagonalΔX={(x,x):x∈X)} is closed in X×X. 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 idX:X→X is continuous, so ΓidX is closed, and ΓidX=ΔX. 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 TheoremThe 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 ParacompactnessThe metric topology is very well-behaved and understood. If one could prove about a topological space (X,T) that there is a metric d on X such that d induces T (in short, if (X,T) is

metrizable), then one immediately gains a long list of useful properties that are met by X. This is why theorems that find conditions on a space which imply metrizability are of interest. The first such result proves thatregularity(separation axiom) and havinga 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 iscountably locally finite.The concept of

local finitenesssounds odd but turns out to be useful. A collection of subsets of X is locally finite if for every point x∈X there is an open set around X which intersects only finitely many of them. The collection of all intervals (n,n+1) is locally finite in R but is not finite. There are also local versions of the properties compactness, metrizability, connectedness, andpath-connectedness. In the latter two cases, neither of the two versions (normal and local) imply the other.Chapter 7: Complete Metric Spaces and Function SpacesThis was the most difficult chapter for me. I find it hard to deal with sets of functions – a function f:R→R can be thought of as a point in the infinite-dimensional space RR, 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 topologyon the space RR corresponds to a sort of "point-wise" study of functions. In particular, a sequence (fn)n∈N of functions converges to a function f 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 theuniform topologyif and only if it converges uniformly (as defined in analysis). Then there is thetopology of compact convergenceand thecompact-open topology.A metric space is called

completeif 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 (X,d) is complete if and only if every sequence of closed nonempty sets A0⊃A1⊃... such that diam(Ai)→0 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 TheoryI'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 GroupThis 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

pathon a topological space (X,T) is a continuous map p:[0,1]→X. The points p(0) and p(1) are called theendpointsof p, and p is said to go from p(0) to p(1). If p(0)=a=p(1), then p is said to be aloopbased at a. The space is path-connected if there exists a path from x to y for any x,y∈X.There is an equivalence relation on the set of all loops based at a fixed point b∈X, where p∼q iff there is a

path homotopybetween them, which can be thought of as a continuous deforming of p into q 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 S1, the path p:s↦e2πis that goes around the circle once is not path homotopic to the path q(s)≡(1,0) 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 p and q where p(1)=q(0) (such as two loops with the same base point) can be connected by simply going along p first and then q. The resulting path is denoted p∗q. This operation can be proven to be well-defined on equivalence classes [p] of paths under homotopy equivalence. Furthermore, for any path p there exists a reverse path ¯¯¯p, and [p∗¯¯¯p]=[c], where c is a constant path. And with that, the set π1(X,b) of all equivalence classes of loops based at b with operation ∗ forms a group! It is called -t-h-e-

a fundamental groupof the space X. Not 'the' because if X is not path connected, then fundamental groups at different base points may be different.My favorite thing about this is that every continuous function f:X→Y with f(b0)=b1 defines a function f∗:π1(X,b0)→π1(Y,b1) via f∗([p])=[f∘p], and the function f∗ is a homomorphism between the two groups π1(X,b0) and π1(Y,b1). That means if f preserves the topology, then f∗ preserves structure! And to make the analogy perfect: if f is a homeomorphism, then f∗ 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 [0,1) and [0,1] 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 R2, there is a single cut that divides both exactly in half.

The fundamental group of S1 is isomorphic to the infinite cyclic group (Z,+). 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 S2 is not homeomorphic to (Z2,+), but to the trivial group ({1},∗). There are also spaces that have non-trivial finite groups.

Chapter 10: Separation Theorems in the PlaneThe last quarter of the book consists of much shorter chapters. I've only started this one, it (among other things) about how continuous maps f:S1→R2 always divide the plane into two regions, one bounded and the other unbounded.

Chapter 11: The Seifert-van Kampen TheoremChapter 12: Classification of Covering SpacesChapter 13: Classification of SurfacesThis 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 bookis that it doesn't seem quite as impressive as

Linear Algebra Done RightandComputability 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.