Compact

From hyperspacewiki
Jump to: navigation, search

Let $(X,\tau)$ be a topological space. We say that a set $\mathcal{O} \subset \tau$ is an open cover of $(X,\tau)$ if $\bigcup \mathcal{O} = X$. A set $\mathcal{O}_0 \subset \mathcal{O}$ is called a subcover of $\mathcal{O}$ if it is also an open cover of $(X,\tau)$. We say that the topological space $(X,\tau)$ is compact if for every open cover $\mathcal{O}$ of $(X,\tau)$ there exists a subcover $\mathcal{O}_0$ whose cardinality is finite.