Countability bases reduce the study of arbitrary open sets to countable families. In metric spaces, every point has a countable local base, but a whole space may or may not have a countable base for all open sets. This distinction leads to first countability, second countability, separability, and Lindelof properties. :::definition[Local Base] Let $(X,d)$ be a metric space and $x\in X$. A family $\mathcal{B}_x$ of open sets containing $x$ is a $local base at x$ if for every open set $G$ containing $x$, there exists $B\in\mathcal{B}_x$ such that $x\in B\subset G$. ::: :::definition[Base for Open Sets] A family $\mathcal{B}$ of nonempty open subsets of $X$ is a $base$ if every open set is a union of members of $\mathcal{B}$. ::: :::definition[Second Countable Metric Space] A metric space is $second countable$ if it has a countable base for its open sets. ::: :::example[Countable Base for the Real Line] The intervals $(a,b)$ with rational endpoints $a,b\in\mathbb{Q}$ and $a<b$ form a countable base for $\mathbb{R}$. ::: :::example[Discrete Metric Spaces] A discrete metric space is second countable if and only if its underlying set is countable. ::: :::theorem[Metric Spaces Are First Countable] Every metric space has a countable local base at each point. In fact, $\{S(x,1/n):n\in\mathbb{N}\}$ is a local base at $x$. ::: :::proof Given that $(X,d)$ is a metric space and $x\in X$. To prove first countability at $x$. Let $G$ be an open set containing $x$. Then there exists $r>0$ such that $S(x,r)\subset G$. Choose $n\in\mathbb{N}$ with $1/n<r$. Then $S(x,1/n)\subset S(x,r)\subset G$. Hence the countable family $\{S(x,1/n)\}$ is a local base at $x$. $\square$ ::: :::solved-problem For $d_1(x,y)=|x|+|x-y|+|y|$ when $x\neq y$ and $d_1(x,x)=0$, find $S_{d_1}(0,\varepsilon)$. ::: :::solution For $x\neq0$, $d_1(0,x)=2|x|$. Hence $$\begin{equation*} S_{d_1}(0,\varepsilon)=\left(-\frac{\varepsilon}{2},\frac{\varepsilon}{2}\right). \end{equation*}$$ $\square$ ::: :::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 Countability Bases in 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/lindelof-and-dense-metric-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/completeness-of-subspaces :::