Let be a disconnected space, say , where and are both open and non-empty. If and , prove that there can be no continuous function such that and .
I want to know HOW important is the "such that" thing.
Here is my attempt:
Let be continuous.
Then are both open in .
Also .
Again . Also hence these are non empty.
The above suggests that is disconnected which is wrong hence we have a contradiction.
Have i missed something or is the proof correct?