Second countable normed vector space

Question

I know, that if $X$ and $Y$ is second countable, then $X_{/Y}$ is second countable.

Is it true, that if $X$ is normed vector space, $Y$ is closed subspace of $X$, and $Y$ is second countable, and also $X_{/Y}$ is second countable, then $X$ is second countable?

My intuition tells me, that answer is no, but I can't find the right example to proof it.

Answer 1

Second countability is equivalent to separability. Given countable dense subsets $A$ of $Y$ and $B$ of $X/Y$ we choose a countable set $C$ of $X$ such that $B=q(C)$, where $q:X\to X/Y$ is the quotient map. Then $A+C=\lbrace a+c: a\in A, c\in C\rbrace$ is countable and dense in $X$.

Indeed, given $x\in X$ and $ \varepsilon>0$ there is $b=q(c)\in B$ such that the quotient norm $$\|q(x)-b\|_{X/Y} = \inf\lbrace \|v\|_X: q(v)=q(x)-b\rbrace <\varepsilon/2.$$ Take such a $v$ with $\|v\|_X <\varepsilon/2$. Then $x-c-v \in kern(q)=Y$ so that there is $a\in A$ with $\|x-c-v-a\|_Y <\varepsilon/2$. This gives $$\|x-(a+c)\|_X \le \|x-c-v-a\|_X+\|v\|_X < \varepsilon.$$