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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Let

and be topological spaces and suppose and are subsets of X such that . Let and be continuous functions such that for all . Then defined byis continuous.

Proof:

Let

beThen

isHence

Therefore

isOffline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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 that

and are closed in as well.*Last edited by JaneFairfax (2008-07-16 20:01:23)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Okay, I think we can say that if

and are either both open or both closed in , then is continuous.If

and are both open in , then and are both open in ; hence is open in .If

and are both closed in , well use this result: is continuous if and only if given any closed subset , is closed in .So if

is 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 .Offline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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