Cont image of connected space is connected proof check?

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SUMMARY

The discussion centers on the proof regarding the connectedness of spaces under a surjective function. The original proof was presented using complex notation, but a participant suggested simplifying it by utilizing the subspace topology on sets A and f(A). This approach leads to a clearer conclusion: if f: X → Y is surjective and X is connected, then Y is also connected. The participant confirmed the validity of the proof and expressed intent to incorporate the suggested simplifications.

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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
 
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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.
 
Thanks so much! I will incorporate this change
 

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