# On Kuratowski partitions

## TL;DR

This paper reviews historical and recent results related to Kuratowski's 1935 problem on the continuity of functions with Baire property preimages, exploring Ellentuck topology and K-ideals.

## Contribution

It compiles and discusses both classical and new findings concerning Kuratowski partitions, especially in the context of Ellentuck topology and K-ideals.

## Key findings

- Connections between Kuratowski partitions and Ellentuck topology.
- Properties of K-ideals related to Kuratowski partitions.
- Summary of historical results and recent advances on Kuratowski's problem.

## Abstract

In 1935 K. Kuratowski posed the problem whether a function f:X ! Y , (X is completely metrizable and Y is metrizable), with the property that a preimage of each open has the Baire property, is continuous apart from a meager set. This paper is a selection of older results related to the question posed by Kuratowski in 1935, coming from among others Solovay and Bukovsk?y, and quite new ones concerning considerations in Ellentuck topology and some propertiesof K-ideals, (i.e. ideals associated with Kuratowski partitions).