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 $+, \cdot : R^{2} \to R$ 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 $z \in R$ 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, $T \in P (P (X))$ , or equivalently $T \subset P (X)$ . 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): $\emptyset, X \in T$

(2): $\forall O_{1}, . . ., O_{n} \in T : (O_{1} \cap . . . \cap O_{n}) \in T$

(3): $\forall {O_{j}}_{j \in J} \subset T : (⋃_{j \in J} O_{j}) \in T$

That is, the topology is closed under arb...