Munkres Topology Article -30 Problem 5: metrizable space with a countable dense set has a countable basis

Question

Show that a metrizable space with a countable dense set has a countable basis.

My try:

Let $X$ be a metrizable space with a countable dense set $D$.

• Consider for each $n\in \Bbb N;B(x,\dfrac{1}{n});x\in X$. Always we have that $B(x,\dfrac{1}{n})\cap D\neq \emptyset $. Let $a\in B(x,\dfrac{1}{n})\cap D$.
• Claim $\scr B$=$\{B(a,\dfrac{1}{n});a\in D;n\in \Bbb N\}$ is a countable basis of $X$.
• Proof:

Let $x$ be arbitrary and $U$ be an open set containing $x$. Then for some positive integer $k$ we have $B(x,\dfrac{1}{k})\subset U$. Then $B(x,\dfrac{1}{2k})\cap D\neq \emptyset $. Let $b\in B(x,\dfrac{1}{2k})\cap D\implies d(x,b)<\dfrac{1}{2k}$.

By Triangle inequality $x\in B(b,\dfrac{1}{2k})\subset B(x,\dfrac{1}{k})\subset U$

• Is the proof correct? Please suggest required edits.

Answer 1

Actually you almost made it. In the end you proved that $x\in U$. This is true since you took $x$ and then $U$ to be an arbitrary open set containing $U$.

What you need to prove is that $$y\in B(b,\frac{1}{2k}) \qquad \Longrightarrow \qquad y\in B(x,\frac{1}{k}).$$ This you can prove by Triangle inequality.