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: $\forall y\in Y\exists Q\left(y\right)\subseteq Y\left[y\text{ is a non-isolated q-point in }Q\right(y\left)\right]$. 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 $f:X\to Y$.