Compactness-like properties for generalized weak topological sums

W. Wistar (William) Comfort

Vol. 60 (1975), No. 1, 31–37

DOI: 10.2140/pjm.1975.60.31

Abstract

It is shown that under suitable conditions (involving cardinal numbers) on a family of spaces {X_i : i ∈ I} with p_i ∈ X_i for i ∈ I, their γ-weak sum {x ∈ Π_i∈IX_i : |{i ∈ I : x_i≠p_i}| < γ} is α-compact in the κ-box topology. For example, there is Corollary 2.5: If α is regular and uncountable and |X_i| < α for all i ∈ I, then the ω-weak sum (= direct sum) is α-compact in the α-box topology; in particular, the direct sum of any set of finite spaces is α-compact in the α-complete topology for regular α > ω.

Mathematical Subject Classification 2000

Primary: 54B10

Milestones

Received: 3 June 1974

Published: 1 September 1975