The question is also posted here.

Let $M=\mathbb{R}$ and $\tau_M=\lbrace U\cup A: U$ open in $\mathbb{R}, A\subset \mathbb{R} \setminus B\rbrace$, where $B$ is a Bernstein set. Then $(M,\tau_M)$ is a topological space called the **Michael Line**. It is a regular Lindelof space.

**Submetrizable** = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.

Let $f: M \to X$ be any one-to-one and onto continuous mapping. Then is $X$ always submetrizable?

Thanks for your help.