Topological space
Set
| definiendum | $\langle X,\mathcal T\rangle \in \mathrm{it} $ |
| postulate | $X,\emptyset\in \mathcal T$ |
| for all | $S\subseteq \mathcal T$ |
| postulate | $\bigcup S\in \mathcal T$ |
| postulate | $S$ … finite $\Rightarrow \bigcap S\in \mathcal T$ |
We call $\mathcal T$ the topology and its elements the open (sub-)sets of $X$.
A comment on the intersection axiom requiring finiteness: A major motivation for topological spaces is $\mathbb R^n$ with the sets “open ball” and in this setting, an infinite intersection of open sets need not be open. E.g. consider the set of open intevals $(-\tfrac{1}{n},\tfrac{1}{n})$.
Reference
Wikipedia: Topological space
Requirements
