Point in Metric Space has Countable Neighborhood Basis
Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Then there exists a basis for the neighborhood system of $a$ which is countable.
Proof
Consider the countable set:
- $\BB_a := \set {\map {B_\epsilon} a: \exists n \in \Z_{>0}: \epsilon = \dfrac 1 n}$
That is, let $\BB_a$ be the set of all open $\epsilon$-balls of $a$ such that $\epsilon$ is of the form $\dfrac 1 n$ for (strictly) positive integral $n$.
Let $N$ be a neighborhood of $a$.
Then by definition of neighborhood:
- $\exists \epsilon' \in \R_{>0}: \map {B_{\epsilon'} } a \subseteq N$
From Between two Real Numbers exists Rational Number:
- $\exists \epsilon \in \Q: 0 < \epsilon < \epsilon'$
Let $\epsilon$ be expressed in canonical form as:
- $\epsilon = \dfrac p q$
where $p$ and $q$ are coprime integers and $q > 0$.
Then:
- $\epsilon := \dfrac 1 q \le \dfrac p q$
Thus $0 < \epsilon < \epsilon'$.
So:
\(\ds x\) | \(\in\) | \(\ds \map {B_\epsilon} a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {x, a}\) | \(<\) | \(\ds \epsilon\) | Definition of Open Ball of Metric Space | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {x, a}\) | \(<\) | \(\ds \epsilon'\) | as $\epsilon < \epsilon'$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \map {B_{\epsilon'} } a\) | Definition of Open Ball of Metric Space | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {B_\epsilon} a\) | \(\subseteq\) | \(\ds \map {B_{\epsilon'} } a\) | Definition of Subset | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {B_\epsilon} a\) | \(\subseteq\) | \(\ds N\) | Subset Relation is Transitive |
From Open Ball is Neighborhood of all Points Inside, $\map {B_\epsilon} a$ is a neighborhood of $a$.
But as $\epsilon = \dfrac 1 q$ where $q \in \Z_{>0}$ it follows that:
- $\map {B_\epsilon} a \in \set {\map {B_\epsilon} a: \exists n \in \Z_{>0}: \epsilon = \dfrac 1 n}$
Hence the result.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Exercise $5$