Although this may be hard to visualize, Alexander's horned sphere is homeomorphic
to a ball. In this case, this means that it can be stretched into a ball without puncturing or breaking it.
Perhaps it is easier to visualize the reverse: stretching the ball into the horned sphere without ripping it. The boundary is, therefore, homeomorphic to a sphere.
I show many more views of this object, provide additional information,
and discuss the artist Cameron Browne in my book
The Möbius Strip:
Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology.
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