GR0568 #65

Forum for the GRE subject test in mathematics.
dumplinghao123
Posts: 24
Joined: Sat Sep 22, 2012 4:34 pm

GR0568 #65

Postby dumplinghao123 » Thu Oct 11, 2012 5:50 pm

Which of the following statements are true about the open interval (0,1) and the closed interval [0,1]?
I. There is a continuous function from (0,1) onto [0,1].
II. There is a continuous function from [0,1] onto (0,1).
III. There is a continuous one-to-one function from (0,1) onto [0,1].

My thoughts:
III is definitely wrong. Because if there is one-to-one function, the topological property of the domain and range should be same.
I is true. There is an example on sfmathgre.blogspot.com

But any thoughts on why II is wrong?

L3inad
Posts: 19
Joined: Thu Oct 11, 2012 7:02 pm

Re: GR0568 #65

Postby L3inad » Thu Oct 11, 2012 7:26 pm

If $K$ is compact and $f$ is a continous function on $K$, then $f(K)$ is going to be compact. In particular you cant map continously then interval [0,1] (which is compact) onto (0,1) (which is not compact).

dumplinghao123
Posts: 24
Joined: Sat Sep 22, 2012 4:34 pm

Re: GR0568 #65

Postby dumplinghao123 » Thu Oct 11, 2012 7:37 pm

L3inad wrote:If $K$ is compact and $f$ is a continous function on $K$, then $f(K)$ is going to be compact. In particular you cant map continously then interval [0,1] (which is compact) onto (0,1) (which is not compact).


got it. Thanks! Very helpful.




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