Connectedness

[Siron]

Let [tex]f: X \to Y[/tex] a continu surjection (onto).
1. If [tex]X[/tex] is connected then [tex]Y[/tex] connected?
2. Suppose [tex]Y[/tex] is connected. When is [tex]X[/tex] connected?

1. This is true, because [tex]f[/tex] is continu
2. I'm wondering, do I have to add a condition to the function [tex]f[/tex] or do I have to give a topology on [tex]Y[/tex] wherefore [tex]X[/tex] is connected? If yes, how?

Thanks in advance!

 

[Siron]

I think I have found the answer. In my opinion the function $f$ has to be open.

 

[Ubistvo]

1. Yes, since $X$ is connected and $f$ continuous, then $f(X)$ is connected, but $f$ is a surjection, so $f(X)=Y$ and $Y$ is connected.
2. As long as $f$ is an homeomorphism.