Letand be topological spaces and suppose and are subsets of X such that . Let and be continuous functions such that for all . Then defined by
Letbe Y-open and consider .
Thenis A-open and is B-open and so there exist X-open sets and such that and .
Thereforeis X-open, showing that is continuous.
I made a mistake in my proof.
What I have is actually, not the other way round.
I need to find an X-open set U such that.
And I may have to assume thatand are closed in as well.
Last edited by JaneFairfax (2008-07-16 20:01:23)
Okay, I think we can say that ifand are either both open or both closed in , then is continuous.
Ifand are both open in , then and are both open in ; hence is open in .
Ifand are both closed in , well use this result: is continuous if and only if given any closed subset , is closed in .
So ifis closed in , is closed in ; where is closed in . Hence is closed in . Similarly (where is closed in ) is closed in . Thus is closed in .