Lindelof Space

Open Cover

Let \left( {X,\tau } \right) 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 for X if X = \bigcup\limits_{\alpha \in I} {{U_\alpha }} .

A sub-collection of an open cover which is itself an open cover is called a sub-cover.

Lindelof Space

A topological space \left( {X,\tau } \right) is said to be a Lindelof space if every open cover of X has a countable sub-cover.

Theorems
• A closed sub-space of a Lindelof space is Lindelof.
• Every second countable space is a Lindelof space.

Lindelof Theorem

Let X be a second countable space. If a non-empty open set G in X is represented as the union of a collection \left\{ {{G_i}} \right\} of open sets, then G can be represented as a countable union of {G_i}'s.