On Subspaces of Replete and Measure Replete Spaces

Abstract

The concepts of repleteness and more generally measure repleteness are investigated for set-theoretic lattices on specific subspaces of a lattice space. These general results are then applied to specific topological spaces, and we obtain as special cases some known theorems as well as some new results concerning for example, ∝-completeness, realcompactness, measure compactness and Borel-measure compactness.

On lattice-perfect measures

“…The terminology and notation are fairly standard and are consistent with those of Wallman [12], Alexandroff [1], Frolik [4], Nobeling [7], Bachman and Sultan [2], as well as Szeto [10] and Grassi [6].…”

On normal lattices and separation and semi‐separation of lattices

“…We begin by reviewing some notation and terminology which is fairly standard (see,for example, Alexsandroff [1], Camacho [2], Grassi [3], and Szeto [4]). We supply some backround and notation for the readers convenience.…”

Outer measures and associated lattice properties

“…[1], [6], [7], [8], [9], [11] (c.f. [10]) that if , 6, M(/-) then there exists a v 6, MR(L) such that/* < v() and/*(X) v(x); if/-is normal and/* 6, I(L), then v 6, IR(L and u is unique.…”