﻿Building off of last doc, we now arrive at topology. What have we got so far? We have numbers with their operators, we have a little bit of category theory with sets as examples. Now what we should really do is provide another example of a category: topological spaces.


What is topology? It’s the study of topological spaces and their structure preserving maps. Remember that the definition of the category is sufficient for defining what “structure preserving” means.


NECESSARY VS SUFFICIENT


Again quick reminder, sufficient means double arrow, necessary single arrow.


So what’s the topology protocol? A topological space is a set X with a topology O, which is a subset of the powerset of X. There are extra restrictions we put on O, such that:
1. Empty set and X are elements
2. Finite intersections of elements are elements
3. Arbitrary unions of elements are elements


Each of the above statements has a reason for being there, but it mostly boils down to “if a is close to b, and b is close to c, then a better be close to c”, and other similar protocol consistency checks.


So when given a specific set X, you now have a choice of topology O to make on it. This choice is in fact the exact choice of choosing what is close to what: ie continuity.


We have some algorithmic choices that have already been figured out for you, so you don’t have to pick individually every subset to put in O every time you want to choose a topology (especially important in infinite sets X, cause picking each subset is unrealistic).


After making a choice of O, you now have a topological space. So what’s a structure preserving map on topological spaces?


Well let’s go through the logic.