On Borel semifilters

TL;DR

This paper characterizes zero-dimensional Borel spaces that are homeomorphic to semifilters, showing they are exactly the homogeneous, non-locally compact spaces, and extends the result to analytic and coanalytic spaces under determinacy assumptions.

Contribution

It provides a complete characterization of Borel spaces homeomorphic to semifilters and extends the result to broader classes under determinacy.

Findings

Zero-dimensional Borel spaces homeomorphic to semifilters are homogeneous and not locally compact.

Under $oldsymbol{ ext{Σ}}^1_1$-Determinacy, the characterization applies to all analytic and coanalytic spaces.

The work builds on and extends previous results by van Engelen and van Mill.

Abstract

Building on work of van Engelen and van Mill, we show that a zero-dimensional Borel space is homeomorphic to a semifilter if and only if it is homogeneous and not locally compact. Under $\mathbf{\Sigma}^1_1$-Determinacy, this result extends to all analytic and coanalytic spaces.