On partitions of Ellentuck-large sets

TL;DR

This paper proves that non-meager subspaces of the Ellentuck space cannot be partitioned into disjoint meager sets with certain Baire property conditions, highlighting a non-metric aspect of Kuratowski partitions.

Contribution

It establishes the non-existence of Kuratowski partitions in non-meager Ellentuck subspaces and shows this is independent of metric considerations.

Findings

Non-meager Ellentuck subspaces lack Kuratowski partitions.

Existence of Kuratowski partitions is not a metric issue.

Remarks on continuous functions in Ellentuck space.

Abstract

It is proved that no non-meager subspace of the space $[\omega]^\omega$ equipped with the Ellentuck topology does admit a Kuratowski partition, that is such a subset cannot be covered by a family $\mathfrak{F}$ of disjoint relatively meager sets such that $\bigcup\mathfrak{F}'$ has the Baire property (relatively) for every subfamily $\mathfrak{F}'\subseteq\mathfrak{F}$. It is also shown that the existence of Kuratowski partitions is not a metric problem. Some remarks concerning continuous restrictions of functions with domain in the Ellentuck space are made.