No homeomorphism (0,1) <--> [0,1]

I'm trying to prove that there isn't a homeomorphism between [0,1]
and (0,1), and I'm stuck. What I know so far is the definitions of
homeomorphism and continuity.

Reductio ad absurdum seems to be the right approach, so:

Let M = [0,1] and M' = (0,1), both with the usual metric.
Let f:M -> M' and let g:M' -> M be bijections,
such that fog = gof = I.

Let f(0) = A and f(1) = B. A != B because f is a bijection, so we
need to deal with two cases: A<B and A>B.

We'll look at the case where A<B:


Try to show that g is not continuous:

Let y be any point in M' such that y>B, and a neighborhood of y, n',
such that n' is a subset of (B,1). Let g(y) = x and let g(n') = n.
In this case, it would seem that n is "missing" some points. Two
problems with this:
1. I can't actually prove that statement. There's no way that
I'm aware of to show that the image of n' doesn't include
all of the points between its GLB and its LUB.
2. Even if this statement was true, it doesn't violate the
definition of continuity as I understand it. My instinct
says that it should, but that doesn't really matter.


Try to show that f is not continuous:

Let x and y be as before, and assume g continuous.

This initially looks more promising. For any delta<1/2, the image
under f of ( 1-2*delta, 1 ) will include points in M' less than B
and points in M' that are greater than B, due to the continuity of
g. So, in theory, all that I have to do is pick a neighborhood n'
as before, and show that the image under f of the open interval
( GLB(g(n')), LUB(g(n')) ) is not a subset of n'.

However, I again encounter a problem. I can't refute the claim that,
with a smaller delta, the neighborhood in M would map to a subset of n'.


Beyond the problems that I've outlined, I have a philosophical concern
that whatever proof I come up with should be generalizable to show
that a disk isn't homeomorphic to a disk with its associated circle,
and to many other cases. What I currently have is dependent upon
concepts like "less than" and "greater than", which don't really
extend.

Hints, anybody?

--
Michael F. Stemper
#include <Standard_Disclaimer>
Outside of a dog, a book is man's best friend.
Inside of a dog, it's too dark to read.

.

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