---- > [!definition] Definition. ([[separable space]]) > A [[topological space]] $X$ is said to be **separable** if it contains a [[countably infinite|countable]] [[dense]] subset, [[closure|that is]], there exists a [[sequence]] $(x_{n})_{n=1}^{\infty}$ of elements of the space such that every nonempty open subset contains at least one element in the sequence. > > Further assuming $X$ is endowed with a [[metrizable|metric]] $d$, an **r-separated subset** of $X$ is a subset $S \subset X$ for which any two distinct points in $S$ are at least $r$ apart: $d(x,y) \geq r$ for all $x \neq y$ in $S$. > [!equivalence] > For [[metrizable]] spaces the notions of separable and [[second-countable space|second-countable]] coincide. ^equivalence > [!basicexample] > - Euclidean space $\mathbb{R}^{n}$ is separable > - Eulicdean [[Lp-norm|Lp spaces]] $L^{p}(E \subset \mathbb{R}^{n})$ [[Euclidean Lp spaces are separable|are separable]] > ^basic-example > [!basicproperties] > - In a separable [[metric space]], every $r$-separated set is [[countably infinite|countable]]. > > > [!proof]- Proof. > > Let $(M,d)$ be separable and $(x_{n})_{n=1}^{\infty}$ be a countable dense subset. [[cover|Cover]] $M$ by the countable family of open balls $\left\{ B_{\frac{r}{2}}(x_{n}) \right\}_{n=1}^{\infty}$. If $S \subset M$ is $r$-separated, then each ball contains at most one point of $S$. Hence $S$ injects into $\mathbb{N}$. ---- #### ---- #### 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 > ```