  We prove that:
  1. If a Hausdorff M-space is a continuous closed image of a submetrizable
space, then it is metrizable.
  2. A dense-in-itself open-closed image of a submetrizable space is
submetrizable if and only if it is functionally Hausdorff and has a countable
pseudocharacter.
  3. Let $Y$ be a dense-in-itself space with the following property: $\forall
y\in Y\ \exists Q(y) \subseteq Y\ [y \text{ is a non-isolated q-point in }
Q(y)]$. If $Y$ is an open-closed image of a submetrizable space, then $Y$ is
submetrizable.
  4. There exist a submetrizable space $X$, a regular hereditarily paracompact
non submetrizable first-countable space $Y$, and an open-closed map $f\colon X
\to Y$.
