On Multi-Split Continuity and Split Homeomorphisms

TL;DR

This paper introduces multi-split continuity as a new, weaker form of continuity in topology, explores its properties, and characterizes split homeomorphisms in compact, regular Hausdorff spaces, linking set-valued analysis and topological deformations.

Contribution

It defines multi-split continuity, studies associated multifunctions, and characterizes split homeomorphisms, connecting set-valued analysis with topological deformations.

Findings

Multi-split continuity generalizes split continuity.

Multi-split continuity is stable under composition.

Split homeomorphisms characterize deformations with cuts and re-glues.

Abstract

We introduce multi-split continuous functions between topological spaces, a weaker form of continuity that generalizes split continuity while being stable under compositions. We will define the associated star multifunction and pre-multi-split multifunctions. Moreover, we will prove that multi-split continuity naturally emerges as the continuity property of selections of finite usco maps, relating their study to set-valued analysis. Finally, we introduce split homeomorphisms and split homeomorphic spaces, showing that for compact, regular Hausdorff spaces, split homeomorphisms characterize deformations with cuts and subsequent re-glues.