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 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.

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