$\mathcal {L}$-realcompactifications as epireflections

Abstract:

If $\mathcal {L}$ is a countably productive normal base on a Tychonoff space $X$, then $\eta (X,\mathcal {L})$ is an ${\mathcal {L}_ \ast }$-realcompact extension of $X$. R. A. Alo and H. L. Shapiro thus generalized the Hewitt realcompactification of $X$. In the following paper, we extend this construction to ${T_1}$-spaces and show that it is an epireflection functor on an appropriate category. We are thus concerned with the question of the extendibility of a continuous map $f:X \to Y$ to a continuous map $g:\eta (X,{\mathcal {L}_X}) \to \eta (Y,{\mathcal {L}_Y})$. We derive necessary and sufficient conditions therefor in the case when ${\mathcal {L}_Y}$ is a nest generated intersection ring on $Y$.