# Connectedness

#### Siron

##### Active member
Let $$f: X \to Y$$ a continu surjection (onto).
1. If $$X$$ is connected then $$Y$$ connected?
2. Suppose $$Y$$ is connected. When is $$X$$ connected?

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

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#### Siron

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

#### Ubistvo

##### New member
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.