Lindelof spaces and dense subsets are two ways countability enters topology. A Lindelof space reduces open covers to countable subcovers. A dense subset approximates every point of the space. These ideas connect directly to separability and second countability in metric spaces. :::definition[Open Cover] A collection $\mathcal{G}$ of open subsets of $X$ is an $open cover$ if $X\subset\bigcup_{G\in\mathcal{G}}G$. ::: :::definition[Subcover] A subcollection $\mathcal{H}\subset\mathcal{G}$ is a $subcover$ if it also covers $X$. ::: :::definition[Lindelof Metric Space] A metric space is $Lindelof$ if every open cover has a countable subcover. ::: :::definition[Dense Subset] A subset $A$ of $X$ is $dense$ in $X$ if $\overline{A}=X$. ::: :::example[Dense Subsets of the Real Line] Both $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$. ::: :::example[Dense Rational Coordinate Points] The set $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$. ::: :::theorem[Second Countable Spaces Are Lindelof] Every second countable metric space is Lindelof. ::: :::proof Given that $X$ has a countable base $\mathcal{B}=\{B_1,B_2,\ldots\}$. To prove that every open cover has a countable subcover. Let $\mathcal{U}$ be an open cover of $X$. For each basis element $B_i$ that is contained in some member of $\mathcal{U}$, choose one such member $U_i$. The chosen family is countable. If $x\in X$, choose $U\in\mathcal{U}$ with $x\in U$. Since $\mathcal{B}$ is a base, there is $B_i$ such that $x\in B_i\subset U$. Therefore the chosen $U_i$ contains $x$. Hence the chosen countable family covers $X$. $\square$ ::: :::solved-problem Prove that $A$ is dense in $X$ if and only if $(X\setminus A)^\circ=\varnothing$. ::: :::solution If $A$ is dense and $X\setminus A$ had nonempty interior, then some nonempty open ball would miss $A$, contradicting $\overline{A}=X$. Conversely, if $(X\setminus A)^\circ=\varnothing$ and $x\notin\overline{A}$, then some ball around $x$ misses $A$, giving a nonempty open subset of $X\setminus A$. This is impossible. Hence $\overline{A}=X$. $\square$ ::: :::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 Lindelof and Dense 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/separable-metric-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/countability-bases-in-metric-spaces :::