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 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 compact space is compact.
• The homeomorphic image of compact space is compact.
• Any continuous bijective function from a compact space X is a hausdorff space Y is a homeomorphism.