Why $\mathbb{R}_\ell$ is first countable?

Question

I have taken some idea from this proof

Question: Why $\mathbb{R}_\ell$ is first countable?

My attempt : Suppose that $\mathcal{B}$ is a uncountable basis for $\mathbb{R}_{\ell}$. Then, given a point $x$ let $B_{x}\in\mathcal{B}$ be a basis element such that $x\in B_{x}\subseteq[x,x+\frac{1}{n})$. Then if $x\neq y$ we have that $B_{x}\neq B_{y}$. but $\inf B_{x}=x$ and $\inf B_{y}=y$.

Now take $y\in B_{x}\subseteq[x,x+\frac{1}{m})$.

We have $\inf B_{x}=\lim_{n\to \infty}B_x$ and $\inf B_{y}=\lim_{n\to \infty}B_y$, then here $\inf B_{x}=\inf B_{y} \space (\because \space \lim_{n\to \infty} \frac{1}{n}= \lim_{m\to \infty} \frac{1}{m})$.

From this we can said that there is no injective function from $\mathbb{R}_{\ell}$ to $\mathcal{B}$ defined by $x\mapsto B_{x}$.

$ \implies\mathcal{B}$ must be countable basis for $\mathbb{R}_{\ell}$.

$ \therefore \mathbb{R}_\ell$ is first countable .

score 2 · Accepted Answer · edited Aug 21 '20 at 06:15

I’m afraid that nothing after Is this proof correct or not makes sense at all. For instance, $\inf B_x=x$, and $\lim_{n\to\infty}B_x$ makes no sense, since $B_x$ is not a function of $n$: $B_x$ is a fixed member of $\mathcal{B}$ that you chose at the very beginning of your argument. Moreover, your conclusion that $\mathcal{B}$ is a countable base for $\Bbb R_\ell$ is simply false: $\Bbb R_\ell$ is not second countable, i.e., it does not have a countable base.

Since $\Bbb R_\ell$ does not have a countable base, you’ll have to show directly that each point of $\Bbb R_\ell$ has a countable local base. That’s easy, and you have the key elements in your answer: for any $x\in\Bbb R_\ell$,

$$\mathcal{B}_x=\left\{\left[x,x+\frac1n\right):n\in\Bbb Z^+\right\}$$

is a local base at $x$: just verify that if $U$ is any open nbhd of $x$, there is some $n\in\Bbb Z^+$ such that $x\in\left[x,x+\frac1n\right)\subseteq U$. Clearly $\mathcal{B}_x$ is countable, so once you’ve done that, you’ve shown that there is a countable local base at each point of $\Bbb R_\ell$, i.e., that $\Bbb R_\ell$ is first countable.

Why $\mathbb{R}_\ell$ is first countable?

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