Cardinal p and a theorem of Pelczynski

TL;DR

This paper explores the consistency and diversity of compactifications of countable discrete spaces with specific remainders, revealing both uniformity and variety depending on the cardinality involved.

Contribution

It demonstrates the conditions under which all such compactifications are homeomorphic and constructs many non-homeomorphic examples based on cardinality.

Findings

Under certain set-theoretic assumptions, all compactifications with a specific remainder are homeomorphic.

There are exponentially many non-homeomorphic compactifications with remainders of continuum cardinality.

The results depend on the set-theoretic context and cardinal characteristics.

Abstract

We show that it is consistent that for some uncountable cardinal k, all compactifications of the countable discrete space with remainders homeomorphic to $D^k$ are homeomorphic to each other. On the other hand, there are $2^c$ pairwise non-homeomorphic compactifications of the countable discrete space with remainders homeomorphic to $D^c$ (where c is the cardinality of the continuum).