Alexander's Horned Sphere
The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere.
It is homeomorphic with the ball 
, and its boundary is therefore a sphere. It is therefore an example of a wild embedding in
. The outer complement of the solid
is not simply connected, and its fundamental
group is not finitely generated. Furthermore,
the set of nonlocally flat ("bad") points of Alexander's horned sphere
is a Cantor set.
The horned sphere as originally drawn by Alexander (1924) is illustrated above.
The complement in 
of the bad points
for Alexander's horned sphere is simply connected,
making it inequivalent to Antoine's horned sphere.
Alexander's horned sphere has an uncountable infinity of wild
points, which are the limits of the sequences of the horned sphere's branch points
(roughly, the "ends" of the horns), since any neighborhood
of a limit contains a horned complex.
A humorous drawing by Simon Fraser (Guy 1983, Schroeder 1991, Albers 1994) depicts mathematician John H. Conway with Alexander's horned sphere growing from his head.
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