Topological space


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 $(-\tfrac{1}{n},\tfrac{1}{n})$.


Wikipedia: Topological space


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