A Continuous Bijection from $\ell^{2}$ Onto a Subset of $\ell^{2}$ Whose Inverse is Everywhere Unboundedly Discontinuous, with an Application to Packing of Balls in $\ell^2$

Abstract

There is a continuous bijection from $ \ell^{2}$ onto a subset of $\ell^{2}$ whose inverse is everywhere unboundedly discontinuous. If $B$ is a ball in $\ell^2$, then the continuous bijection defined on $\ell^2$ maps countably many mutually disjoint balls of $\ell^2$ into countably many mutually disjoint balls in $B$, making those images mutually disjoint.