On Multi-Split Continuity and Split Homeomorphisms

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 define the associated star multifunction and pre-multi-split multifunctions. Moreover, we 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 Hausdorff spaces, split homeomorphisms characterize deformations with cuts and subsequent re-glues.