Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2008-07-16 04:01:41

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

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.

Offline

#2 2008-07-16 15:01:07

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

Re: Pasting lemma

I made a mistake in my proof. swear

What I have is actually

, not the other way round. 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 (2008-07-16 20:01:23)


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

A: Click here for answer.

Offline

#3 2008-07-16 20:01:05

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

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
, we’ll 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.

Offline

#4 2008-07-16 20:39:57

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

Re: Pasting lemma




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

A: Click here for answer.

Offline

Board footer

Powered by FluxBB