GR0568 #65

[26186514]

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]

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).

[26186514]

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.