A separable metric space is one that can be approximated everywhere by a countable set. This is a powerful condition because many uncountable spaces in analysis, such as $\mathbb{R}$, $\mathbb{R}^n$, $C[0,1]$, and $\ell^p$, are separable. Separability links density, countability bases, and Lindelof properties. :::definition[Separable Metric Space] A metric space $(X,d)$ is $separable$ if it contains a countable dense subset, that is, there is a countable $A\subset X$ such that $\overline{A}=X$. ::: :::example[Standard Separable Spaces] The spaces $\mathbb{R}$, $\mathbb{R}^n$, $C[0,1]$, and $\ell^p$ for $1\leq p<\infty$ are separable. ::: :::example[Inseparability of Bounded Sequences] The space of bounded real sequences with the supremum metric is not separable, because it contains uncountably many $0$-$1$ sequences separated by distance $1$. ::: :::theorem[Equivalent Countability Conditions] For a metric space $(X,d)$, the following are equivalent: $X$ is separable, $X$ is second countable, and $X$ is Lindelof. ::: :::proof Given that $(X,d)$ is a metric space. To prove the equivalence. If $X$ is separable, let $D=\{x_1,x_2,\ldots\}$ be countable and dense. The balls $S(x_i,r)$ with $r\in\mathbb{Q}^+$ form a countable base, so $X$ is second countable. If $X$ is second countable, then it is Lindelof by the second countable Lindelof theorem. If $X$ is Lindelof, then for each $n$, the cover $\{S(x,1/n):x\in X\}$ has a countable subcover $\{S(x_{n,k},1/n):k\in\mathbb{N}\}$. The countable set $D=\{x_{n,k}:n,k\in\mathbb{N}\}$ is dense, because every point is within $1/n$ of some member of $D$ for each $n$. Hence $X$ is separable. $\square$ ::: :::solved-problem Prove that every subspace of a separable metric space is separable. ::: :::solution Let $D=\{x_1,x_2,\ldots\}$ be dense in $X$ and let $Y\subset X$. For each $n,m$, choose one point from $Y\cap S(x_n,1/m)$ when this set is nonempty. The chosen set is countable and dense in $Y$ by the triangle inequality. $\square$ ::: :::exercise [1] State the main definition from this lesson. [2] State the main theorem from this lesson. [3] Give one example illustrating the theorem. [4] Explain one common mistake related to this topic. ::: :::answer [1] The main definition is the first titled definition in the lesson. [2] The main theorem is the titled theorem proved in the lesson. [3] The examples in the lesson provide standard choices. [4] A common mistake is to ignore the metric or the ambient space when applying the definition. ::: :::faq[Frequently Asked Questions] Q: Why is the ambient metric space important? A: Openness, closedness, closure, and boundary are interpreted relative to the chosen metric space. Q: Can the same set behave differently in different metric spaces? A: Yes. Changing the metric or the ambient space may change topological behaviour. Q: How should students check a definition? A: They should return to balls, closures, and sequences, depending on the context. ::: :::call-to-action[Continue from Separable Metric Spaces] subtitle: Review the definitions and proofs before moving through the metric topology sequence. button: Next Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/category-and-nowhere-dense-sets button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/lindelof-and-dense-metric-spaces :::