| definiendum | $\langle X,\mathcal{T}\rangle \in\mathrm{it} $ |
| inclusion | $\langle X,\mathcal{T}\rangle$ … topological space |
| for all | $\mathcal{T'}\subseteq \mathcal{T}$ |
| for all | $K\subseteq \bigcup \mathcal{T}'$ |
| exists | $\mathcal{T}''\subseteq \mathcal{T}'$ |
| postulate | $K\subseteq\bigcup \mathcal{T}''$ |
| postulate | $\mathcal{T}''$ … finite |
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 refined 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.