Closed and open-closed images of submetrizable spaces

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 $f\colon X \to Y$.