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.

• 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 $${G_i}’s$$.