ℒ-realcompactifications as epireflections

Abstract

If if is a countably productive normal base on a Tychonoff space X, then r¡(X,-a*) is an-Sf^-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 ^-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/: X-+Y\.o a continuous map g:r¡(X,¿£'x)->-rj(Y,J?'r). We derive necessary and sufficient conditions therefor in the case when Sfr is a nest generated intersection ring on Y. All of the topological spaces which we consider are assumed to satisfy the Fj-separation axiom. A family =5? of closed subsets of a space X is called a countably productive separating base if the following are satisfied: (i) J? is a ring, i.e. it is closed under finite unions and finite intersections and 0,Xe^. (ii) ¿£ is countably productive, i..e closed under countable intersections. (iii) =Sf is separating, i.e. whenever x e X-S, S closed in X, there exist Lx, L2eJ¡? such that x e Lx, 5c L2, and LxC\L2=0 (E. F. Steiner [9]). If in addition to the above properties, (iv) =S? is nest generated, i.e. L e =£? implies the existence of En, Fn e 3?, neN, such that Fn+1^X-Ena Fn and L=f) {Fn:n e N} (A. K. Steiner and E. F. Steiner [8], [10]), then J? is called a nest generated intersection ring. (R. A. Alo and H. L. Shapiro [2] use the term strong delta normal base.) In [8], A. K. Steiner and E. F. Steiner prove that a nest generated intersection ring =Sf is normal, i.e. Lx, L2 e^C, and LxC\L2=0 implies the existence of L'x, L!2e££ such that LxnL[=0, L2C\L'2=0, and L'XKJL'2 = X. A well-known example of a nest generated intersection ring is Z(X), the family of all zero sets of a Tychonoff space X.