A homeomorphism on $s$ not conjugate to an extendable homeomorphism

Abstract:

Consider $s = \Pi _{i = 1}^\infty {( - 1,1)_i}$ and its compactification $Q = \Pi _{i = 1}^\infty {\left [ { - 1,1} \right ]_i}$. Anderson and Bing asked whether for every homeomorphism $f:s \to s$ there is a homeomorphism $\phi :s \to s$ such that ${\phi ^{ - 1}}f\phi$ is extendable to a homeomorphism $\overline {{\phi ^{ - 1}}f\phi :\;} Q \to Q$. The aim of this note is to construct a counterexample to this question.