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#1 2008-07-17 02:01:41

JaneFairfax
Legendary Member

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Pasting lemma

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 by



is continuous.

Proof:

Let
be Y-open and consider
.

Then
is A-open and
is B-open and so there exist X-open sets
and
such that
and
.

Hence



Therefore
is X-open, showing that
is continuous. smile


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#2 2008-07-17 13:01:07

JaneFairfax
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Re: Pasting lemma

I made a mistake in my proof. swear

What I have is actually

, not the other way round. http://i41.photobucket.com/albums/e271/Carduelis_carduelis/BangHead.gif

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 (2008-07-17 18:01:23)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#3 2008-07-17 18:01:05

JaneFairfax
Legendary Member

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Re: Pasting lemma

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
.


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#4 2008-07-17 18:39:57

JaneFairfax
Legendary Member

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Re: Pasting lemma










Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

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