Extending homeomorphisms on Cantor cubes

TL;DR

This paper investigates conditions under which homeomorphisms between closed subsets of Cantor cubes can be extended to the entire space, focusing on the preservation of certain interior properties related to cardinals.

Contribution

It establishes that homeomorphisms preserving $ ext{lambda}$-interiors can be extended to autohomeomorphisms of the entire Cantor cube $D^ au$, generalizing previous extension results.

Findings

Homeomorphisms preserving $ ext{lambda}$-interiors can be extended to the whole space.

Extension is possible for any cardinal $ ext{lambda}$ under the given conditions.

Provides a criterion for extending homeomorphisms in Cantor cubes.

Abstract

We discuss the question of extending homeomorphism between closed subsets of the Cantor discontinuum $D^\tau$. It is established that any homeomorphism $f$ between two closed subsets of $D^\tau$ can be extended to an autohomeomorphism of $D^\tau$ provided $f$ preserves the $\lambda$-interiors of the sets for every cardinal $\lambda$.