# 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.