A space is realcompact if its a closed subspace of an arbitrary product of real lines, with product topology. Its known that a discrete space is realcompact if and only if it has non-measurable cardinality. The proof of this result basically follows trivially from definitions. Its also known that...

Yes, its true. Note the following result from the article The sup = max problem for the extent and the Lindelöf degree of generalized metric spaces, II by Hirata: Theorem. (corollary 2.3) Let $X$ be a semi-stratifiable space with $e(X) = \kappa$, where $\text{cf}(\kappa) > \omega$. Assume that $\...

I know that a discrete space is realcompact iff its non-measurable and I've been able to prove that the same holds for metrizable spaces here using the following result of Hirata. Theorem. (corollary 2.3) Let $X$ be a semi-stratifiable space with $e(X) = \kappa$, where $\text{cf}(\kappa) > \omeg...