On the convergence and character spectra of compact spaces

Abstract

This implies the existence of 0-dimensional compact T 2 spaces whose cardinality and weight spectra omit lots of singular cardinals of "small" cofinality.

Reflecting character and pseudocharacter

“…Arguments made in [13] show that, if X is a compact Hausdorff space, and κ ∈ χS(X), then {κ, cf(κ)} ⊆ cS(X).…”

“…One of this results is used in Section 3 to obtain more equivalences with CH, extending Theorem 1.4. In Section 4 we use the theory of character and convergence spectra developed in [13] to make several remarks that give some partial answers to Problem 1.8.…”

“…As far as we know, the only partial answer in the literature is: To investigate this issue, we will use some concepts defined in [13].…”

“…Theorem 4.10 ( [13]). If ̺ is an infinite cardinal with ̺ = cf(̺) > ω, and X is a compactum with F (X) > ̺, then ̺ ∈ χS(X).…”

“…Definition 4.9 (see [13]). Given a space X, F (X) is the smallest cardinal κ for which there is no free sequence of size κ in X.…”