$\mathfrak{c}$-many types of a $\Psi$-space

TL;DR

This paper constructs a large family of $ ext{ extcursive c}$-many $ ext{ extcursive c}$-sized almost disjoint families with pairwise non-homeomorphic $ ext{ extcursive extPsi}$-spaces, including Luzin and branch families, for certain cardinals.

Contribution

It demonstrates the existence of $ ext{ extcursive c}$-many non-homeomorphic $ ext{ extcursive extPsi}$-spaces derived from different almost disjoint families under specific cardinal conditions.

Findings

Existence of $ ext{ extcursive c}$-many non-homeomorphic $ ext{ extcursive extPsi}$-spaces

Construction includes Luzin families and branch families of $2^ ext{ extcursive c}$

Applicable for cardinals $ ext{ extcursive c}$ with cofinality greater than $ ext{ extcursive extcursive c}$

Abstract

We show that for any cardinal $\omega<\kappa \leq \mathfrak{c}$ with $cf(\kappa) > \omega$, there are $\mathfrak{c}$ many AD families whose $\Psi$-spaces are pairwise non-homeomorphic and they can be Luzin families or branch families of $2^\omega$.