Attaining the spread at cardinals of cofinality ω

Kenneth Kunen and Judith Roitman

Vol. 70 (1977), No. 1, 199–205

DOI: 10.2140/pjm.1977.70.199

Abstract

Let λ be a singular cardinal of cofinality ω. We investigate the question: does every Hausdorff space with spread λ have a discrete subspace of cardinality λ? The answer is “yes” if λ > 2^ℵ₀ or if λ < 2^ℵ₀ and MA holds; however, for λ < 2^ℵ₀ an answer of “no” is consistent with the axioms of set theory. The proof involves showing the equivalence of the question with one about category in the real line. Similar results hold for the width of a space.

Mathematical Subject Classification

Primary: 02K30, 02K30

Milestones

Received: 14 January 1976

Published: 1 May 1977