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!!!
question about topology. Need help!!!

 Posts: 27
 Joined: Sun Oct 17, 2010 4:57 am
Re: question about topology. Need help!!!
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!!!
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 preimage 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
=> 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 preimage 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

 Posts: 15
 Joined: Sat May 14, 2011 6:24 am
Re: question about topology. Need help!!!
Thats clearly defined.
Keep it up
Keep it up