Topological Space

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 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).

Here we introduce several more basic definitions so that we can talk about them in other articles.