Axiomatizing topology through continuous maps
Solution 1:
A topology space on $X$, is a set of functions $X \to \{False, True\}$, such that.
- Constant functions are continuous.
- For any collection of continuous functions $f, g, h \dots$ (possibly infinite), the function $z(x) = f(x) \vee g(x) \vee h(x) \vee \dots$ is continuous.
- For any two continuous functions $f$ and $g$, $z(x) = f(x) \wedge g(x)$ is continuous.
(Interesting note: If you let functions be potential computations, $False$ represent infinite loops, and $True$ termination, something interesting happens.)