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

Arbitrary union, Arbitrary intersection

Topological space

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Topological space

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