# Compact Space

Cover and Sub-Cover

Let $X$ be a topological space. A collection $\left\{ {{A_\alpha }:\alpha \in I} \right\}$ of subsets of $X$ is said to be a cover of $X$ if $X = \bigcup\limits_{\alpha \in I} {{A_\alpha }}$.
A sub-collection of $\left\{ {{A_\alpha }:\alpha \in I} \right\}$ is said to be a sub-cover of $X$ if itself is a cover of $X$.

Open Cover and Open Sub-Cover

Let $X$ be a topological space. A collection $\left\{ {{U_\alpha }:\alpha \in I} \right\}$ of open subsets of $X$ is said to be an open cover of $X$ if $X = \bigcup\limits_{\alpha \in I} {{U_\alpha }}$.
A sub-collection of $\left\{ {{U_\alpha }:\alpha \in I} \right\}$ is said to be a sub-cover of $X$ if itself is an open cover of $X$.

Compact Space

A compact space is a topological space in which every open cover has a finite sub-cover.

Compact Subspace

A compact subspace of a topological space is a subspace which is compact as a topological space in its own right.

Examples:
• Every finite topological space is compact.
• Let $\left( {X,\tau } \right)$ be a topological space where $\tau$ consists of a finite number of elements, then $X$ is a compact space.

Theorems
• An infinite set with co-finite topology is a compact space.
• The real line $\mathbb{R}$ is not compact.
• Every closed subspace of a compact space is compact.
• The continuous image of a compact space is compact.
• The homeomorphic image of a compact space is compact.
• Any continuous bijective function from a compact space $X$ is a hausdorff space, $Y$ is a homeomorphism.