An embedding characterization of almost realcompact spaces

Abstract:

Any Hausdorff space X can be embedded into the product space $\Pi \{ R_f^ + :f \in {C_ + }(X)\}$, where ${R^ + }$ is the set of all nonnegative reals with the topology consisting of ${R^ + }$ and all sets of the form $\{ x \in {R^ + }:x < a\} ,a \in R$, and ${C_ + }(X)$ is the set of all continuous functions from X to ${R^ + }$. Almost realcompact Hausdorff spaces are characterized as maximal Hausdorff subspaces in their closures in the product.