**"Extending homeomorphisms on Cantor cubes"**
#### Valov, VeskoThis 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 $f$. |