Remarks on the Ostrovsky's Theorem

TL;DR

This paper refines Ostrovsky's Theorem by establishing the necessity of the one-to-one condition for continuous open-resolvable mappings and demonstrates that the open-LCn function problem has a negative answer for all n > 1.

Contribution

It proves the necessity of the one-to-one condition in Ostrovsky's Theorem and shows the open-LCn function problem is negatively resolved for all n > 1.

Findings

The one-to-one condition is necessary in Ostrovsky's Theorem.

The open-LCn function problem has a negative solution for all n > 1.

Provides counterexamples or proofs related to open-LCn functions.

Abstract

In this paper we prove that the condition one-to-one of continuous open-resolvable mapping is necessary in the Ostrovsky's Theorem. Also we get that the Ostrovsky's Problem (Is every open-LCn function between Polish spaces piecewise open for n = 2, 3, ... ?) has a negative solution for any n > 1.