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 ${G_i}’s$.