Paranormal spaces under $\diamondsuit ^{*}$

Abstract

Abstract. We prove that paranormal spaces of character ≤ ω 1 are ω 1 -collectionwise Hausdorff assuming the set-theoretic principle ♦ * . This gives an affirmative answer to problem 197 in Problems I wish I could solve, by W. S. Watson (Open Problems in Topology (1990), 37-76).

Applying Set Theory

“…Fleissner implicitly used a stronger combinatorial principle ♦ SS . In 1979, the author and independently both Fleissner and Alan D. Taylor all saw (as mentioned in [12,13] that a weaker principle, ♦ * ω 1 , would suffice. Later Smith and Szeptycki [12] derive better results.…”

“…In 1979, the author and independently both Fleissner and Alan D. Taylor all saw (as mentioned in [12,13] that a weaker principle, ♦ * ω 1 , would suffice. Later Smith and Szeptycki [12] derive better results. On more recent results on diamond and strong negation, see [14] and references there.…”