If a product of topological spaces is metrizable is it true that every topological space that constitutes the product is metrizable?

Question

If a product of topological spaces is metrizable is it true that every topological space is metrizable? If not,could someone provide me with an example?

Answer 1

If $Y$ is not empty, say $y\in Y$, and $d$ is a metric on $X\times Y$, then consider $(x,x')\mapsto d((x,y),(x',y))$

Answer 2

Every factor $X_i$ of a non-empty product space $\prod_{i \in I} X_i$ is topologically a subspace of that product. So every factor will inherit all hereditary topological properties (like metrisability and Hausdorffness, regularity, and many others).