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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 $\Rightarrow \bigcap S\in \mathcal T$ |
Discussion
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 $(-\tfrac{1}{n},\tfrac{1}{n})$.
Predicates
A topological space is compact if each set $K$ which is covered by some collection of open sets, can in fact be covered by a finite collection of open sets. I'd say that if topology is about collections of elements, then compact spaces are the ones which are of small enough cardinality so that you can reason with finitely running enumerations/computations.
Formally: Let $X$ by a set with topology $\mathcal T$.
| predicate | $X$ … compact $\equiv \forall (\mathcal{T'}\subseteq \mathcal{T}).\ \forall (K\subset X).\ (K\subseteq \bigcup \mathcal{T}' \implies\ \exists (\mathcal{T}''\subseteq \mathcal{T}').\ \mathrm{finite}(\mathcal{T}'')\land K\subseteq\bigcup \mathcal{T}'')$ |
Reference
Wikipedia: Topological space
Parents
Requirements
