Spaces without remote points

Eric Karel van Douwen and Jan van Mill

Vol. 105 (1983), No. 1, 69–75

DOI: 10.2140/pjm.1983.105.69

Abstract

All spaces considered are completely regular and X^∗ denotes βX − X. The point x ∈ X^∗ is called a remote point of X if x∉Cl_βXA for each nowhere dense subset A of X. If y ∈ Y , then the space Y is said to be extremally disconnected at y if y∉Ū∩V whenever U and V are disjoint open sets. In this paper we construct two noncompact σ-compact spaces X, one locally compact and one nowhere locally compact, such that X has no remote points, and in fact such that βX is not extremally disconnected at any point.

Mathematical Subject Classification 2000

Primary: 54D35

Secondary: 54D40, 54G20

Milestones

Received: 11 April 1978

Revised: 10 March 1982

Published: 1 March 1983