On a problem of Juhasz and van Mill

TL;DR

This paper addresses a longstanding open problem in topology by showing that the existence of a universal bound for dense in itself subsets in certain compact spaces is false under weaker conditions, specifically for Hausdorff and sequentially compact spaces.

Contribution

It provides a negative answer to a modified version of Juhasz and van Mill's problem, expanding understanding of dense subsets in compact topological spaces.

Findings

No universal bound exists for dense in itself subsets in Hausdorff, sequentially compact spaces.

The problem remains open under original conditions of regularity and countable compactness.

The result clarifies limitations in the structure of dense subsets in compact topologies.

Abstract

A 27 years old and still open problem of Juhasz and van Mill asks whether there exists a cardinal kappa such that every regular dense in itself countably compact space has a dense in itself subset of cardinality at most kappa. We give a negative answer for the analogous question where_regular_ is weakened to_Hausdorff_, and_coutnably compact_ is strengthened to_sequentially compact_.