Document Type
Lecture
Publication Date
9-21-2005
Abstract
A topological space �� is said to be compact if every filter base on �� has a cluster point (Vietoris, 1920) or, equivalently, every open cover of �� has a finite subcover (Alexandrov and Urysohn, 1922). Many other notions of compactness arise in topology and analysis, including countably compact, sequentially compact, paracompact, metacompact, Eberlein compact, angelic, pseudocompact, and feebly compact spaces. If �� ⊆ ��, then �� is a compact (in any sense) subset of �� whenever �� as a subspace of �� is compact. Let �� now be a family of subsets of ��. What does it mean for �� to be compact? Can we have a common principle that covers many of these definitions and applies to families of subsets? I believe I now know the answers. I will discuss the notions of P = R-compact (midcompact, ultracompact) at a family �� of sets and the role of filter ��-compactness as a unifying scheme.
Relational Format
presentation
Recommended Citation
Labuda, Iwo, "Compact Families of Sets" (2005). Analysis Seminar. 48.
https://egrove.olemiss.edu/math_analysis/48
