A Simple Example of a Lindelöf Space

A Lindelöf space is a topological space in which every open cover has a finite or countable subcover. A standard, useful example is the real line $\mathbb{R}$ with its usual topology. Its Lindelöf property follows from the fact that it has a countable basis.

The real line is Lindelöf

Consider any open cover $\mathcal{U}$ of $\mathbb{R}$. This means that every member of $\mathcal{U}$ is open and that their union is all of $\mathbb{R}$. We will find a countable collection of members of $\mathcal{U}$ that still covers the line.

The intervals $(p,q)$ with rational endpoints $p<q$ form a countable basis for the usual topology on $\mathbb{R}$. In particular, if $x$ belongs to an open set $U$, there is a rational-endpoint interval $(p,q)$ such that

$$x\in(p,q)\subseteq U.$$

Now collect all rational-endpoint intervals that are contained in at least one member of the cover:

$$\mathcal{B}=\{(p,q):p,q\in\mathbb{Q},\ p<q,\text{ and }(p,q)\subseteq U\text{ for some }U\in\mathcal{U}\}.$$

There are only countably many rational-endpoint intervals, so $\mathcal{B}$ is countable. For each interval $B\in\mathcal{B}$, choose one cover member $U_B\in\mathcal{U}$ containing it. The collection of these chosen cover members is countable.

It covers $\mathbb{R}$: given any $x\in\mathbb{R}$, choose a cover member $U$ containing $x$. Since $U$ is open, some rational-endpoint interval $B$ satisfies $x\in B\subseteq U$. This interval belongs to $\mathcal{B}$, and the selected cover member $U_B$ contains $B$, hence contains $x$. Therefore the selected members form a countable subcover.

This argument proves the general theorem that every second-countable space is Lindelöf. A space is second-countable when it has a countable basis for its topology.

A shorter example: any countable space

There is an even simpler way to produce examples. Every countable topological space is Lindelöf, regardless of its topology. Given an open cover, choose one cover member containing each point. Since there are only countably many points, these choices give a countable subcover.

For instance, the rational numbers $\mathbb{Q}$ with the topology inherited from $\mathbb{R}$ are Lindelöf because $\mathbb{Q}$ is countable. The real line is often a more instructive example because it is uncountable, yet still Lindelöf by second countability.

Lindelöf does not mean compact

Compactness requires every open cover to have a finite subcover. Lindelöfness permits a countably infinite subcover, so it is a weaker condition in general.

For example, the open sets

$$U_n=(-n,n),\qquad n=1,2,3,\ldots,$$

cover $\mathbb{R}$. No finite selection covers the whole line: among finitely many such intervals, the largest one is still bounded. Thus $\mathbb{R}$ is not compact, even though it is Lindelöf.

Every compact space is Lindelöf, because a finite subcover is also finite or countable. The reverse implication does not hold, as the real-line example shows.

Common mistakes

• Requiring a finite subcover: That is the definition of compactness, not Lindelöfness. A Lindelöf space may need an infinite countable subcover.
• Assuming separation axioms are part of the definition: Lindelöfness itself does not require a space to be Hausdorff or regular. Some theorems about Lindelöf spaces add such hypotheses, but the definition does not.
• Thinking every uncountable space fails the property: The real line is uncountable and Lindelöf. Countability of the space is one sufficient condition; a countable basis is another.
• Using an uncountable discrete space as a Lindelöf example: An uncountable discrete space is not Lindelöf. Its cover by singleton sets has no countable subcover.

So, if you need a familiar uncountable example, use $\mathbb{R}$ with its usual topology. If you only need the simplest possible example, any countable space—such as $\mathbb{Q}$—will do.