# nLab Lindelöf topological space

Contents

Named after Ernst Leonard Lindelöf.

# Contents

## Definition

A topological space is a Lindelöf space if every open cover has a countable sub-cover.

## Properties

regular Lindelöf spaces are paracompact

### Implications

• a metric space has a $\sigma$-locally discrete base

• Nagata-Smirnov metrization theorem

• a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of countable cover of $X$.

• second-countable spaces are separable: use the axiom of countable choice to choose a point in each set of countable cover.

• separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover.

• Lindelöf spaces are trivially also weakly Lindelöf.

• a space with a $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base.

• a first-countable space is obviously Fréchet-Urysohn.

• a Fréchet-Uryson space is obviously sequential.

• a sequential space is obviously countably tight.

## References

Last revised on April 5, 2019 at 19:55:59. See the history of this page for a list of all contributions to it.