Cover and Sub-Cover:
            Let X be a topological space. A collection  of subsets of X is said to be a cover of X if .
            A sub-collection of   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  of open subsets of X is said to be an open cover of X if .
            A sub-collection of   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  be a topological space where  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  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.