“If a topological space X is second countable, then every open cover of X has a countable subcover (i.e., X is Lindelöf).”

Theorem 30.3. Suppose X has a countable basis. Then: (a) Every open covering of X contains a countable subcover. (b) There exists a countable subset of X that is dense in X. Note. A topological space for which every open cover has a countable subcover is often called a Lindelöf space. Since R is second-countable then R is a Lindelöf space, by Theorem 30.3(a).

Using countable choice, then: Every second-countable topological space is Lindelöf, i.e. any open cover admits a countable subcover. Let be a countable base of the topology. Given any open cover of , we can form the index set of those that are contained in some . By assumption . The axiom of countable choice provides now a section of .

### Property-theoretic statement The property of topological spaces of being a second-countable space implies, or is stronger than, the property of being a Lindelof space. ### Verbal statement Any second-countable space is Lindelof. ... A topological space is termed **second-countable** if it admits a countable basis. A topological space is termed **Lindelof** if every open cover of the space has a countable subcover.

Second-countability implies Lindelöf in all topological spaces, regardless of whether they are metric spaces or not. ... ### Sufficient Condition We have from Second-Countable Space is Lindelöf that second-countability implies Lindelöf in all topological spaces, regardless of whether they are metric spaces or not. ... Hence the result, by definition of second-countable space.

Let X be a second-countable topological space with countable basis B = {B_n}. Let U be an open cover of X. For each B_n in B, choose U_n in U such that B_n ⊆ U_n whenever possible. Then {U_n : n in N} is a countable subcover of U that still covers X. Hence every second-countable space is Lindelöf.

“A topological space X is called Lindelöf if every open cover of X has a countable subcover. … The Lindelöf theorem, stating that every second countable space is Lindelöf, was proved by him for Euclidean spaces as early as 1903 in [Lin03].”

“(b) Let X be a metrizable Lindelöf space. For each n ∈ Z+, the collection of open sets {B(x, 1/n) | x ∈ X} covers X. Pick a countable subcover and call it Bn. Then the collection B = ⋃∞_{n=1} Bn is a countable basis.” This exercise is the converse direction: it uses the assumption that **every open cover of X has a countable subcover (Lindelöf)** in a metrizable space to prove that X has a countable basis (i.e. is second countable), showing the close relationship between Lindelöf and second countability.

The question asks whether every Lindelöf space is second countable. Multiple answers point out that this is false in general: there are Lindelöf spaces that are not second countable. The discussion, however, takes as known the standard result that every second countable space is Lindelöf but emphasizes that the converse implication fails.

9. Lindelöf space: every open covering has a countable subcovering. 11. Second countability axiom: has a countable basis for its topology; is said to be second-countable. 12. A second countable space is both Lindelöf and separable. 13. If a metric space is Lindelöf or separable then it is second countable.

A topological space X is called second countable if it has a countable base. It is called Lindelöf if every open cover of X has a countable subcover. It is a standard theorem that second countable spaces are Lindelöf; given an open cover, one uses the countable base to extract a countable subfamily still covering X.

At around 1208–1244 seconds the lecturer states: “Let’s suppose we’ve got a second countable topological space X. Then **every open cover of X has a countable subcover**… that is the definition of what it means for a space to be Lindelöf… So we’ll use that terminology sometimes as well.” Later (around 1322–1384 seconds) they explain the proof by constructing, from a countable basis, a countable subcollection of the given open cover that still covers X.

In standard general topology texts (e.g., Munkres, *Topology*, 2nd ed., Section on second countability and separability), it is proved that if a topological space X has a **countable basis** (is second countable), then for any open cover of X one can select, for each basis element, an open set from the cover containing it. Because the basis is countable, this yields a **countable** subcollection of the original cover which still covers X. This result is usually summarized as: “Every second countable space is Lindelöf.”

Second-Countable is a property of a topological space whose structure can be defined by a countable basis consisting of a countable collection of open sets. This condition belongs to the field of topology and guarantees that a space is also first-countable, separable, and Lindelöf. ... Second-countability is a powerful property that guarantees a space is also first-countable, separable (contains a countable dense subset), and Lindelöf (every open cover has a countable subcover).

The question asks: "Why does second countable imply Lindelöf?" One of the posted answers explains: If X has a countable base {B_n}, and {U_i} is an open cover of X, then for each n there is some U_i(n) containing B_n. The countable subcollection {U_i(n)} still covers X, so any second countable space is Lindelöf.

The question asks whether the axiom of countable choice is required to prove that second-countable spaces are Lindelöf. Several answers explain that in ZF the usual textbook proof that every second-countable space is Lindelöf does not require any form of the axiom of choice beyond what is built into ZF, while other constructive frameworks may explicitly invoke countable choice. Thus, the theorem 'every second-countable space is Lindelöf' is accepted in standard treatments, though its dependence on choice can be discussed in weaker foundations.

This question asks for an example of a Lindelöf space that is not second countable. The accepted answer gives such an example, showing that the Lindelöf property does not imply second countability. The result still assumes the standard theorem that second countable implies Lindelöf, but highlights that the converse is refuted by the example.

One highly upvoted answer notes that there are many spaces that are Lindelöf but not second countable, illustrating that **Lindelöf does not imply second countable**. The discussion, however, treats as standard the fact that “every second countable space is Lindelöf” and focuses instead on constructing counterexamples to the converse.

A space is a Lindelof space if every open cover of X has a countable subcover, i.e., a countable subcollection of that cover that is also a cover of X. The Euclidean spaces R and R^n are both Lindelof and separable. ... Whenever a space has a countable base (i.e. it is a second countable space), the two notions [Lindelöf and separable] coincide. For example, all Euclidean spaces are second countable.

The question asks whether the statement “every second countable space is Lindelöf” holds in ZF (set theory without the Axiom of Choice). One answer explains that **in ZF alone the implication can fail**, and that standard proofs of ‘second countable ⇒ Lindelöf’ use some form of countable choice. Thus, while in ordinary mathematics (ZF + AC or at least countable choice) second countable spaces are Lindelöf, in choice-less settings this need not be provable.

A topological space is second-countable if it has a base for its topology consisting of a countable set of subsets. ... Lindelöf: every open cover has a countable sub-cover. ... For topological spaces, second-countable implies Lindelöf and separable.

In ZF, the following statements are considered: (i) Every second countable metric space is separable. (ii) Every second countable metric space has a countable base consisting of ... It follows that the properties being second countable and Lindelöf are independent. In particular, the statement 'every second countable metric space is Lindelöf' is not provable in ZF without additional choice principles.

This video proves the theorem: Show that every second countable space is Lindelof space. The statement discussed is that in a second countable topological space, every open cover admits a countable subcover, so the space is Lindelöf.

The video is titled: “if a space is second countable then prove that every open cover of it has a countable subcover.” Around the beginning, the instructor recalls: “a topological space is said to be second countable if it has a countable base.” The rest of the lecture is devoted to proving that from any open cover of such a space, one can extract a **countable subcover**, thereby establishing that second countable spaces are Lindelöf.

No. Being Lindelöf does not imply second-countable. There are standard counterexamples of Lindelöf spaces that are not second-countable (for instance, certain uncountable product spaces with the box topology). However, the implication in the other direction holds: every second-countable space is Lindelöf, but not conversely.