Solution of a problem due to Bing

It is proved in this article that for Alexander's “horned” sphere $S_A^2$ in $E^3$ there exists a pseudoisotopy $F_t$ of the space $E^3$ onto itself which transforms the boundary of the three-dimensional simplex $\sigma^3$ in $S_A^2$ such that the continuous mapping $F_1$ has a countable set of nondegenerate preimages of points each of which is not a locally connected continuum in $E^3$ intersecting $\partial\sigma^3$ in a singleton.
This answers affirmatively a question posed by R. H. Bing in the Mathematical Congress in Moscow in 1966.