:PROPERTIES:
:ID:       b0784577-9691-4c8e-a8e4-974a7c9c4949
:ROAM_ALIASES: "topological space" "open set" topology
:END:
#+title: Topological Space
#+author: Preston Pan
#+description: Algebraic? Geometric? Fantastic!
#+options: broken-links:t

* Definition
A topological space is a set $X$, equipped with a topology. That is, it is equipped with a collection of subsets that are considered to be the /open
sets/ of that topology. These open sets must obey several rules:
1. $\cup_{\alpha \in A}U_{\alpha}$ is open, if all $U_{\alpha}$ are open.
2. $\cap_{n=0}^{N}U_{n}$ is open, if $N$ is finite and $U_{n}$ are open.
3. $\emptyset$ is open, and $X$ is open.
the [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][dual]] concept to open sets are closed sets, which are the complements of open sets. Note that closed sets can also be open sets, and vise versa; a
simple example is the space itself, in any topology; $X$ is open by definition, yet it is also closed because $\emptyset^{c} = X$. This is not just a trivial
example; these "clopen" sets are fairly common (this is in fact the terminology people use).
* More Basic Definitions
Here we introduce several more basic definitions so that we can talk about them in other articles.
** Closure
:PROPERTIES:
:ID:       1954ee72-ffce-4586-ad8a-a46c39c8f77d
:ROAM_ALIASES: interior closure
:END:
The /closure/ of a set $F$ in a topological space $X$ is denoted $\overline{F}$ and is defined as the smallest closed set which contains every open set
$U \subset F$. Likewise, the /interior/ of a set is defined as the largest open set which is inside $F$.