FOM: Re: precursors of Cantor
Matthew Frank
mfrank at math.uchicago.edu
Fri Jun 14 11:42:27 EDT 2002
On Fri, 14 Jun 2002, charles silver wrote:
> What would happen if a one-to-one correspondence were defined as a
> one-to-one map in both directions instead of being one-to-one plus onto in
> only one direction?
This would work, at least classically, but it still misses the categorical
point: we want f:S->T and g:T->S such that fg is the identity on T and gf
is the identity on S.
There's a neat example in topology that is fairly parallel. We define
topological spaces S and T to be homeomorphic if there are continuous maps
f and g satisfying the above condition. It is not enough to require a
bijective continuous map from S to T and a bijective continuous map from T
to S:
Let S = an open annulus,
plus one point on the inner circle
and a closed arc on the outer circle
Let T = an open rectangle,
plus the closure of the left side
and the closure of the left half of the top
There is a bijective continuous map from S to T
by shrinking the hole until it becomes a point.
There is a bijective continuous map from T to S
by connecting the left and right sides of the rectangle.
The spaces are not homeomorphic.
This is what goes wrong if you don't insist on two-sided inverses.
--Matt
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