## question about topology. Need help!!!

Forum for the GRE subject test in mathematics.
dragoonlp
Posts: 54
Joined: Thu Mar 25, 2010 4:38 pm

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

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!!!

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.

Napot
Posts: 9
Joined: Sat Apr 09, 2011 10:03 pm

### 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 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

johnrichard
Posts: 15
Joined: Sat May 14, 2011 6:24 am

### Re: question about topology. Need help!!!

Thats clearly defined.

Keep it up 