FOM: Re: precursors of Cantor

[Matthew Frank]

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