Is the canonical projection $p: X \times X \rightarrow X$ a covering map of the topological space $X$?

Question

Is the canonical projection $p: X \times X \rightarrow X$ a covering map of the topological space $X$? I would say it satisfies the definition.

For instance: $p: \mathbb{R}^2 \rightarrow \mathbb{R}$.

Answer 1

It's not unless $X$ is discrete. Note that each fibre $X\simeq p^{-1}(a)$ has to be discrete for it to be a covering map, or put in another way, $p$ is in general not a local homeomorphism.