Cont image of connected space is connected proof check?

[dumbQuestion]

I was wondering if someone can tell me if my approach to the proof is a correct one. (rather than typing it all out here and making a mess of the notation, I typed it up in latex and did a screencap then put that on imgur, so the following link has the proof)

http://i.imgur.com/gNFToKx.png

 

[micromass]

It's a good proof.

You could have simplified a lot of the notation by working with the subspace topology on $A$ and $f(A)$. This would have simplified the question to: let $f:X\rightarrow Y$ be surjective, then if $X$ is connected then $Y$ is connected. The proof should then be the same as you gave, but the notation would be easier.

 

[dumbQuestion]

Thanks so much! I will incorporate this change