Can topology be studied without $\mathbb{R}$?

Question

My friend is a programmer who works with countable math objects (like countable graphs, or groups). He often reads the professional literature where a lot of topological concepts are mentioned. That's why he's trying to study topology on his own. Alas, in those textbooks, there are a lot of examples concerning $\mathbb{R}$. Thus, since he usually works with countable objects, he starts thinking that not much of the concepts would be helpful. Can he omit the examples with $\mathbb{R}$ without loss of generality? Are there general or algebraic topology textbooks where all such examples are omitted, i.e. they are concentrated on finitistic objects?

Answer 1

Yes, it is possible, and very useful to build an intuition of topology without $\mathbb R$. It is possible to view topology as capturing data about semi-decidability. If your friend knows Haskell (it's a very math-influenced purely functional programming language), ask them to read Martín Escardó's so-called Barbados notes, number 46 in https://www.cs.bham.ac.uk/~mhe/papers/index.html, AKA "Synthetic topology of data types and classical spaces".

The general idea is that the open sets in the topology are those properties which we can decide are true in finite time, but cannot decide if they are false in finite time.

To expand, the topology on $\mathbb R$ allows us to semi-decide if a point $p$ belongs to some interval $I$ when we are able to access the point $p$ one digit at a time. For example, let's pick $I = (0, 1)$.

• Now, if we see a point $p = 0.1\dots$ we know immediately that this point is in $I$, so we can answer that $p \in_? I = \texttt{TRUE}$.
• If we see a point $p = 2.\dots$, we know immediately that this point $p$ is not in $I$, so we can answer $p \in_? I = \texttt{FALSE}$
• Now consider the point $p = 0.9999\dots$ When do we decide that this point $p$ does not belong to $I$?
  • we can't decide after seeing two digits $(0.9\dots)$, for the point $p$ could be $0.98$ and it would then belong to $I$
  • we can't decide after seeing three digits ($0.99\dots)$, for the point $p$ could be $0.998$ and it would then belong to $I$
  • in general, we would need to read the entire number $0.999\dots$ to decide that $p \not \in I$. Clearly, this takes infinite time!

So, if $p \in I$, then we will always answer $p \in_? I = \texttt{TRUE}$ in finite time. But if $p \not \in I$, then we may not answer $p \in_? I = \texttt{FALSE}$ in finite time. Hence, the relation $\in_?$ is semi-dedicable, and the topology of $\mathbb R$ precisely captures the semi-decidable nature of membership in an interval.

More broadly speaking, discrete topology is vital to computer science, since finite topology is the study of preorders and partial orders. This is precisely the notion that is necessary to assign mathematical meaning to computer programs, through a branch of mathematics called as denotational semantics

I've written some blog posts about these issues that may be of some help to get started, since I collect links as well:

• Topology is really about computation
• Finite topologies and DFS numbers