Compact space

Let $(X, \tau)$ - a topological-space.

It is called compact, if $$ \forall {G_{\alpha}: G_{\alpha} \in \tau}{A}:, \bigcup{\alpha \in A} G_{\alpha} \supset X \implies \exists {G_1, \ldots, G_K}{k=1}^{K} \hookrightarrow \bigcup{k=1}^{K} G_{k} \supset X. $$

Recursive definition

• $(X, \tau)$ - compact $\iff$ each closed-set in $X$ is compact

In hausdorff-space

If $(X, \tau)$ - a hausdorff-space then each compact-space is closed-set.

Proposition

If metric-space is a compact-space then it is separable-space.