<s>
The	O
Alexander	B-Algorithm
horned	I-Algorithm
sphere	I-Algorithm
is	O
a	O
pathological	O
object	O
in	O
topology	B-Architecture
discovered	O
by	O
.	O
</s>
<s>
The	O
Alexander	B-Algorithm
horned	I-Algorithm
sphere	I-Algorithm
is	O
the	O
particular	O
embedding	O
of	O
a	O
sphere	O
in	O
3-dimensional	O
Euclidean	O
space	O
obtained	O
by	O
the	O
following	O
construction	O
,	O
starting	O
with	O
a	O
standard	O
torus	O
:	O
</s>
<s>
The	O
horned	B-Algorithm
sphere	I-Algorithm
,	O
together	O
with	O
its	O
inside	O
,	O
is	O
a	O
topological	O
3-ball	O
,	O
the	O
Alexander	B-Algorithm
horned	I-Algorithm
ball	I-Algorithm
,	O
and	O
so	O
is	O
simply	O
connected	O
;	O
i.e.	O
,	O
every	O
loop	O
can	O
be	O
shrunk	O
to	O
a	O
point	O
while	O
staying	O
inside	O
.	O
</s>
<s>
The	O
exterior	O
is	O
not	O
simply	O
connected	O
,	O
unlike	O
the	O
exterior	O
of	O
the	O
usual	O
round	O
sphere	O
;	O
a	O
loop	O
linking	O
a	O
torus	O
in	O
the	O
above	O
construction	O
cannot	O
be	O
shrunk	O
to	O
a	O
point	O
without	O
touching	O
the	O
horned	B-Algorithm
sphere	I-Algorithm
.	O
</s>
<s>
This	O
is	O
one	O
of	O
the	O
earliest	O
examples	O
where	O
the	O
need	O
for	O
distinction	O
between	O
the	O
categories	O
of	O
topological	O
manifolds	B-Architecture
,	O
differentiable	O
manifolds	B-Architecture
,	O
and	O
piecewise	O
linear	O
manifolds	B-Architecture
became	O
apparent	O
.	O
</s>
<s>
Now	O
consider	O
Alexander	B-Algorithm
's	I-Algorithm
horned	I-Algorithm
sphere	I-Algorithm
as	O
an	O
embedding	O
into	O
the	O
3-sphere	O
,	O
considered	O
as	O
the	O
one-point	O
compactification	O
of	O
the	O
3-dimensional	O
Euclidean	O
space	O
R3	O
.	O
</s>
<s>
The	O
closure	O
of	O
the	O
non-simply	O
connected	O
domain	O
is	O
called	O
the	O
solid	B-Algorithm
Alexander	I-Algorithm
horned	I-Algorithm
sphere	I-Algorithm
.	O
</s>
<s>
Although	O
the	O
solid	O
horned	B-Algorithm
sphere	I-Algorithm
is	O
not	O
a	O
manifold	B-Architecture
,	O
R	O
.	O
H	O
.	O
Bing	O
showed	O
that	O
its	O
double	O
(	O
which	O
is	O
the	O
3-manifold	O
obtained	O
by	O
gluing	O
two	O
copies	O
of	O
the	O
horned	B-Algorithm
sphere	I-Algorithm
together	O
along	O
the	O
corresponding	O
points	O
of	O
their	O
boundaries	O
)	O
is	O
in	O
fact	O
the	O
3-sphere	O
.	O
</s>
<s>
One	O
can	O
consider	O
other	O
gluings	O
of	O
the	O
solid	O
horned	B-Algorithm
sphere	I-Algorithm
to	O
a	O
copy	O
of	O
itself	O
,	O
arising	O
from	O
different	O
homeomorphisms	O
of	O
the	O
boundary	O
sphere	O
to	O
itself	O
.	O
</s>
<s>
The	O
solid	B-Algorithm
Alexander	I-Algorithm
horned	I-Algorithm
sphere	I-Algorithm
is	O
an	O
example	O
of	O
a	O
crumpled	O
cube	O
;	O
i.e.	O
,	O
a	O
closed	O
complementary	O
domain	O
of	O
the	O
embedding	O
of	O
a	O
2-sphere	O
into	O
the	O
3-sphere	O
.	O
</s>
<s>
One	O
can	O
generalize	O
Alexander	O
's	O
construction	O
to	O
generate	O
other	O
horned	B-Algorithm
spheres	I-Algorithm
by	O
increasing	O
the	O
number	O
of	O
horns	O
at	O
each	O
stage	O
of	O
Alexander	O
's	O
construction	O
or	O
considering	O
the	O
analogous	O
construction	O
in	O
higher	O
dimensions	O
.	O
</s>
<s>
Another	O
example	O
,	O
also	O
found	O
by	O
Alexander	O
,	O
is	O
Antoine	O
's	O
horned	B-Algorithm
sphere	I-Algorithm
,	O
which	O
is	O
based	O
on	O
Antoine	O
's	O
necklace	O
,	O
a	O
pathological	O
embedding	O
of	O
the	O
Cantor	O
set	O
into	O
the	O
3-sphere	O
.	O
</s>
