# Compact Space

**Cover and Sub-Cover**

Let be a topological space. A collection of subsets of is said to be a cover of if .

A sub-collection of is said to be a sub-cover of if itself is a cover of .

**Open Cover and Open Sub-Cover**

Let be a topological space. A collection of open subsets of is said to be an open cover of if .

A sub-collection of is said to be a sub-cover of if itself is an open cover of .

**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 be a topological space where consists of a finite number of elements, then is a compact space.

**Theorems**

• An infinite set with co-finite topology is a compact space.

• The real line 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 is a hausdorff space, is a homeomorphism.