| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
I am not sure if this is a standard theorem, but I proved the following
while solving a problem:
Let X be a topological space (make it second countable if you like).
Let A be a compact subsete of X and f:X-->Y be continuous and 1-1.
Show that f restrict to A is a homeomorphism, i.e. f:A-->f(A) is a
homeomorphism.
Eugene
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1361.1 | point set topology is so fascinating | GUESS::DERAMO | Dan D'Eramo | Sat Dec 29 1990 15:52 | 17 |
That's a standard result, if you add that Y is Hausdorff.
A is compact, so its closed subsets are compact.
The continuous image of a compact set is compact.
Compact subsets of a Hausdorff space are closed.
So the image of a closed subset of A under f is also
closed. Thus (f|A)-inverse is continuous and so f|A
is a homeomorphism.
For a counterexample when Y is not Hausdorff, let X = A =
{0,1} with the discrete topology (every subset open), let
Y be {0,1} with the indiscrete topology (only the empty
set and the entire set are open), and let f be the
identity function.
Dan
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