36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

"Extending homeomorphisms on Cantor cubes"

Valov, Vesko

This is a joint paper with E. Shchepin. 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 $\lambda$. This is a non-metrizable analogue of the Ryl-Nardjewski theorem stating that if $X$ is a proper closed subset of the Cantor set $D^{\aleph_0}$ and $f$ is a homeomorphism of $X$ onto $f(X)$ such that $f(\rm{int}X) =\rm{int}~f(X)$, then there exists an autohomeomorphism of $D^{\aleph_0}$ extending $f$.

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