Which topological properties are retained by Product Space $X\times X$

Question

Given a space $X$ with some properties--say, for example, Hausdorff, compact, metrizable, etc.--does $X\times X$ retain all topological properties (forgive me for imprecise language, as I'm unsure of proper terminology)?

If not, what are some examples of properties not necessarily retained?

If so, how do we know this? Do we have to verify one-by-one, per property, or is there some generalized proof? For things such as Hausdorff and compact, I can think of proofs, but two properties is a very small subset of all possible properties.

Answer 1

• If $X$ is normal (or $T_4$) $X \times X$ need not be normal.
• If $X$ is Lindelöf, $X \times X$ need not be.
• If $X$ is countably compact, $X \times X$ need not be (contrasting with the preservation of compactness, even for arbitrary products).
• If $X$ is ccc, $X \times X$ need not be.

What does get preserved: connectedness, local compactness, local connectednes, (local) path-connectedness, separation axioms $T_0$ to $T_{3\frac12}$, separability, second and first countability, to name some elementary ones.

The first need some counterexamples, and for the positive ones: we have to check them separately and many are straightforward to check.