Realcompactness in terms of inverse limit of regular Lindelöf spaces

Question

$X$ is a realcompact iff it's a limit of an inverse system of regular Lindelöf spaces

I don't have any idea how to do this. Any help is appreciated!

Answer 1

One direction is not hard: if $X$ is such a limit of spaces $X_\sigma, \sigma \in S$, then $X$ is a closed subspace of $\prod_\sigma X_\sigma$ (Prop. 2.5.1) and all $X_\sigma$ are realcompact (being regular Lindelöf (Thm 3.11.12)) and realcompactness is preserved both by products and closed subspaces (3.11.4; 3.11.5) and so $X$ is realcompact (which is already 3.11.6, where this is stated separately as a corollary). Or shorthand: directly from 3.11.6 plus 3.11.12.

If $X$ is realcompact, $X \subseteq \Bbb R^I$ (closed subset) for some index set $I$, WLOG. Then apply example 2.5.3. but with the set $S:=[I]^{\le \aleph_0}$ of at most countable subsets of $I$ instead. Then 2.5.6 implies that the sets $X_J=\pi_J[X]$, the "traces" of $X$ in the subpower $\Bbb R^J$, where $J \in S$, form an inverse system with limit $X$ and all $X_J$ are Lindelöf regular (as all subspaces of $\Bbb R^{\aleph_0}$ are).

It's really not that hard given the results in the text so far; that's why Engelking writes "observe" and not "show" or "prove" (which are the next levels of difficulty in his exercises). He's rather careful to make sure that such exercises do not contain really new ideas but are applications of results/methods and examples shown thus far in the book.