# Lindelof Space

**Open Cover**

Let be a topological space. A collection of open subsets of is said to be an open cover for if .

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

**Lindelof Space**

A topological space is said to be a Lindelof space if every open cover of 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 be a second countable space. If a non-empty open set in is represented as the union of a collection of open sets, then can be represented as a countable union of .