$X,Y$ be metric spaces , $f:X \to Y$ be continuous and injective ; if $f(X)$ is separable , then is $X$ also separable?

Question

Let $X,Y$ be metric spaces , $f:X \to Y$ be continuous and injective ; if $f(X)$ is separable , then is it true that $X$ is also separable ? I only know that the continuous image of a separable space is separable . Please help . Thanks in advance

Answer 1

Take $X = (\mathbb{R}, \tau_{\text{discrete}})$ and $Y = (\mathbb{R}, \tau_{\text{std}})$. Both are metric spaces and the identity function is continuous and injective (bijective). But $X$ is not separable, while $Y = f(X)$ is.