Covers

Definition: Cover

Let $X$ be a set.

A collection $\mathcal{C}$ of subsets of $X$ is a cover of $X$ iff $X$ is a subset of the union of $\mathcal{C}$.

$$X\subseteq \bigcup \mathcal{C}$$

Definition: Subcover

Let $X$ be a set and let $\mathcal{C}$ be a cover of $X$.

A subcover of $\mathcal{C}$ is subcollection of $\mathcal{C}$ which is still a cover of $X$.

Definition: Open Cover

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

A cover of $X$ is an open cover iff all of its elements are open sets.

Compactness

Definition: Compactness of a Topological Space

A topological space is compact iff each of its open covers has a finite subcover.

Note: Compactness of a Subset

A subset of a topological space is called compact iff it is compact as a subspace.