Countably compact extensions and cardinal characteristics of the continuum

TL;DR

The paper investigates when regular first-countable spaces admit dense embeddings into regular first-countable countably compact (and related) extensions and shows these embedding properties are tightly linked to the values of continuum cardinal characteristics. By combining transfinite embedding constructions with non-embedding results, it proves equivalences among $\omega_1=\mathfrak c$, $\mathfrak b=\mathfrak c$, and $\mathfrak b=\mathfrak s=\mathfrak c$, and extends these to zero-dimensional and pseudocompact contexts. It also analyzes the role of $P_{\mathfrak c}$-points and Nyikos spaces under PFA, establishing that normal Nyikos spaces become compact while non-normal cases admit rigid embeddings. Overall, the work reveals deep connections between compact-like extensions in topology and cardinal characteristics of the continuum, with implications for the structure of Nyikos spaces and related embeddings.

Abstract

In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak b=\mathfrak s=\mathfrak c$ if and only if every regular first-countable space of weight $< \mathfrak c$ can be densely embedded into a regular first-countable countably compact space.

Theorems & Definitions (62)

• Theorem 1.2
• Theorem 1.4: Bardyla, Zdomskyy
• Theorem 1.5: Bardyla, Zdomskyy
• Theorem 1.7: Terada, Terasawa
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6