Lindelöf space

Definition

Let $X$ be a topological space. The space $X$ is a Lindelöf space if every open cover $\mathcal{U}$ of $X$ has a countable subcover. That is, there is a countable family $\mathcal{V} \subseteq \mathcal{U}$ with $\bigcup \mathcal{V} = X$ .

Remarks

Example: $\R$ with the usual topology is Lindelöf. It is second-countable, with the countable base of open intervals with rational end points. For an open cover $\mathcal{U}$ , pick for each basic interval contained in some member of $\mathcal{U}$ one such member; these countably many members cover $\R$ .

Non-example: an uncountable set $X$ with the discrete topology is not Lindelöf. The cover $\set{\set{x} : x \in X}$ is an open cover, and every subcover must contain all of its uncountably many members.

The Sorgenfrey line is Lindelöf, but its square, the Sorgenfrey plane, is not. So a product of two Lindelöf spaces need not be Lindelöf.

Reading path

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Open cover concept
Topological space concept
Lindelöf space definition

Tag	`ETDW4AK1W1`
URI	`m2://ETDW4AK1W1`
Kind	definition
Writing	W0 stub
MSC	54D20
Defines	Lindelöf space
Uses	Topological space, Open cover
Changed	2 November 2026

Lindelöf space

Definition

Remarks

Reading path

Further reading

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