---- > [!definition] Definition. ([[second-countable space]]) > If a [[topological space]] $X$ has a [[countably infinite|countable]] [[topology generated by a basis|basis generating its topology]], the $X$ is said to satisfy the **second countability axiom**, or be **second-countable**. > [!basicexample] > - $\mathbb{R}$ has as countable [[basis for a topology|basis]] the collection of [[open interval|open intervals]] with [[rational]] endpoints. > - $\mathbb{R}^{n}$ has as countable [[basis for a topology|basis]] the collection of all [[product topology|products]] of [[open interval]]s having rational endpoints. > - $\mathbb{R}^{\mathbb{N}}$ has as countable basis the collection of all products $\prod_{n \in \mathbb{N}}^{}U_{n}$, where $U_{n}$ is an [[open interval]] with rational end points for finitely many values of $n$, and $U_{n}=\mathbb{R}$ for all other values of $n$. > Each result above follows from the [[basis-nestling characterization of comparing topologies]], coupled with the [[the rationals are dense in the reals|density of the rationals in the reals]]. \ More generally: Any [[metric space]] $(X,d)$ with a countable dense subset $D \subset X$ is [[second-countable space|second-countable]]. > [!basicproperties] > - Products of [[second-countable space]]s are [[second-countable space|second-countable]] > - Subspaces of [[second-countable space|second-countable spaces]] are [[second-countable space|second-countable]] > [!basicnonexample] > The second countability axiom is stronger than the [[first-countable space|first]]. It is so strong, in fact, that not every [[metric space|metric space]] satisfies it. In the [[uniform topology]], $\mathbb{R}^{\mathbb{N}}$ satisfies the first axiom but not the second. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```