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question about topology. Need help!!!

Posted: Thu Apr 07, 2011 11:21 pm
by dragoonlp
My question is:

If a function f: X -> Y is continuous and bijective.

Then the statement:
"If X is compact, then Y is compact."

Is true or not?
If not, could you give me a example?

Thx a lot!!!

Re: question about topology. Need help!!!

Posted: Thu Apr 07, 2011 11:40 pm
by fireandgladstone
It's true. The image of a compact space under a continuous function is compact. The function is bijective so the image is the codomain.

Re: question about topology. Need help!!!

Posted: Mon Apr 11, 2011 9:34 pm
by Napot
Let { V_alpha} alpha in A be a open cover of Y
=> Union over all alpha of V_alpha contains Y
=> the preimage of the Union over all alpha of V_alpha contains X
=>The Union over all alpha of the preimage of V_alpha contains X
By definition of open cover, each V_alpha is open and since f is continuous each the pre-image of V_alpha is open
=>X has a finite subcover since X is compact. Union from i = 1 to n of U_i contains X.
=> f( Union from i = 1 to n of U) contains Y
=> Y has a finite subcover for an arbitrary open cover
=>Y is Compact

Re: question about topology. Need help!!!

Posted: Sat May 14, 2011 6:42 am
by johnrichard
Thats clearly defined.

Keep it up :)