|
|
| Author |
Message |
mina_world
science forum Guru Wannabe
Joined: 20 Jul 2005
Posts: 186
|
 Posted: Wed Jul 19, 2006 4:55 am Post subject:
topology with connected.
|
|
|
hello sir~
if A is a connected subset of X,
int(A), bd(A) is connected ?
---------------------------------
um... i think..
if A = (1,2)U[2,3) in usual topology of R.
A is connected.
and
int(A) = (1,2)U(2,3) is not connected.
bd(A) = {1,2,3} is not connected.
is this right ? |
|
| Back to top |
|
 |
William Elliot
science forum Guru
Joined: 24 Mar 2005
Posts: 1906
|
| Posted: Wed Jul 19, 2006 6:09 am Post subject:
Re: topology with connected.
|
|
|
On Wed, 19 Jul 2006, mina_world wrote:
| Quote: | if A is a connected subset of X,
int(A), bd(A) is connected ?
if A = (1,2)U[2,3) in usual topology of R.
A is connected.
and
int(A) = (1,2)U(2,3) is not connected.
bd(A) = {1,2,3} is not connected.
is this right ?
No. A = (1,3) = int A |
bd A = { 1,3 }
Find a planar example for connected A, disconnected int A |
|
| Back to top |
|
|
The World Wide Wade
science forum Guru
Joined: 24 Mar 2005
Posts: 790
|
| Posted: Wed Jul 19, 2006 6:26 am Post subject:
Re: topology with connected.
|
|
|
In article <e9kds2$4is$1@news2.kornet.net>,
"mina_world" <mina_world@hanmail.net> wrote:
| Quote: | hello sir~
if A is a connected subset of X,
int(A), bd(A) is connected ?
---------------------------------
um... i think..
if A = (1,2)U[2,3) in usual topology of R.
A is connected.
and
int(A) = (1,2)U(2,3) is not connected.
bd(A) = {1,2,3} is not connected.
is this right ?
|
No. A = (1,3). So int(A) = A. bd(A) = {1,3}, so that's a step in
the right direction. Note that if A is a connected subset of R,
then int(A) is always connected. Proof: A is an interval.
To find a connected A such that int(A) is not connected, think
about subsets of R^2. (Try to find one so that both int(A) and
bd(A) are not connected.) |
|
| Back to top |
|
|
mina_world
science forum Guru Wannabe
Joined: 20 Jul 2005
Posts: 186
|
| Posted: Wed Jul 19, 2006 7:09 am Post subject:
Re: topology with connected.
|
|
|
"The World Wide Wade" <waderameyxiii@comcast.remove13.net> wrote in message
news:waderameyxiii-7C1A4B.23264818072006@comcast.dca.giganews.com...
| Quote: | In article <e9kds2$4is$1@news2.kornet.net>,
"mina_world" <mina_world@hanmail.net> wrote:
hello sir~
if A is a connected subset of X,
int(A), bd(A) is connected ?
---------------------------------
um... i think..
if A = (1,2)U[2,3) in usual topology of R.
A is connected.
and
int(A) = (1,2)U(2,3) is not connected.
bd(A) = {1,2,3} is not connected.
is this right ?
No. A = (1,3). So int(A) = A. bd(A) = {1,3}, so that's a step in
the right direction. Note that if A is a connected subset of R,
then int(A) is always connected. Proof: A is an interval.
To find a connected A such that int(A) is not connected, think
about subsets of R^2. (Try to find one so that both int(A) and
bd(A) are not connected.)
|
um... i think...
X = {a,b,c,d,e}
T = {X,empty,{a,b,c},{a,b},{c}}
A = {a,b,c,d} is connected.
int(A) = {a,b,c} is not connected.
in the case R^2,
A = {R^2 - (Q*{0})}
int A = R^2 - (R*{0})} is not connected.
but i can't find the example with both int(A) and bd(A). |
|
| Back to top |
|
|
toni.lassila@gmail.com
science forum beginner
Joined: 18 May 2006
Posts: 2
|
| Posted: Wed Jul 19, 2006 9:44 am Post subject:
Re: topology with connected.
|
|
|
mina_world wrote:
| Quote: | "The World Wide Wade" <waderameyxiii@comcast.remove13.net> wrote in message
news:waderameyxiii-7C1A4B.23264818072006@comcast.dca.giganews.com...
In article <e9kds2$4is$1@news2.kornet.net>,
"mina_world" <mina_world@hanmail.net> wrote:
hello sir~
if A is a connected subset of X,
int(A), bd(A) is connected ?
---------------------------------
um... i think..
if A = (1,2)U[2,3) in usual topology of R.
A is connected.
and
int(A) = (1,2)U(2,3) is not connected.
bd(A) = {1,2,3} is not connected.
is this right ?
No. A = (1,3). So int(A) = A. bd(A) = {1,3}, so that's a step in
the right direction. Note that if A is a connected subset of R,
then int(A) is always connected. Proof: A is an interval.
To find a connected A such that int(A) is not connected, think
about subsets of R^2. (Try to find one so that both int(A) and
bd(A) are not connected.)
um... i think...
X = {a,b,c,d,e}
T = {X,empty,{a,b,c},{a,b},{c}}
A = {a,b,c,d} is connected.
int(A) = {a,b,c} is not connected.
in the case R^2,
A = {R^2 - (Q*{0})}
int A = R^2 - (R*{0})} is not connected.
but i can't find the example with both int(A) and bd(A).
|
Take two disjoint infinite closed strips { (x,y) in R^2 : a <= x <= b,
y in R } and join them with a line segment in such a way that they
connect only in a boundary point. |
|
| Back to top |
|
|
mina_world
science forum Guru Wannabe
Joined: 20 Jul 2005
Posts: 186
|
| Posted: Wed Jul 19, 2006 10:06 am Post subject:
Re: topology with connected.
|
|
|
toni.lassila@gmail.com 작성:
| Quote: | mina_world wrote:
"The World Wide Wade" <waderameyxiii@comcast.remove13.net> wrote in message
news:waderameyxiii-7C1A4B.23264818072006@comcast.dca.giganews.com...
In article <e9kds2$4is$1@news2.kornet.net>,
"mina_world" <mina_world@hanmail.net> wrote:
hello sir~
if A is a connected subset of X,
int(A), bd(A) is connected ?
---------------------------------
um... i think..
if A = (1,2)U[2,3) in usual topology of R.
A is connected.
and
int(A) = (1,2)U(2,3) is not connected.
bd(A) = {1,2,3} is not connected.
is this right ?
No. A = (1,3). So int(A) = A. bd(A) = {1,3}, so that's a step in
the right direction. Note that if A is a connected subset of R,
then int(A) is always connected. Proof: A is an interval.
To find a connected A such that int(A) is not connected, think
about subsets of R^2. (Try to find one so that both int(A) and
bd(A) are not connected.)
um... i think...
X = {a,b,c,d,e}
T = {X,empty,{a,b,c},{a,b},{c}}
A = {a,b,c,d} is connected.
int(A) = {a,b,c} is not connected.
in the case R^2,
A = {R^2 - (Q*{0})}
int A = R^2 - (R*{0})} is not connected.
but i can't find the example with both int(A) and bd(A).
Take two disjoint infinite closed strips { (x,y) in R^2 : a <= x <= b,
y in R } and join them with a line segment in such a way that they
connect only in a boundary point.
|
yes, maybe this is a form of | |--| |
thank you very much. |
|
| Back to top |
|
|
William Elliot
science forum Guru
Joined: 24 Mar 2005
Posts: 1906
|
| Posted: Wed Jul 19, 2006 11:02 am Post subject:
Re: topology with connected.
|
|
|
On Wed, 19 Jul 2006 mina_world@hanmail.net wrote:
| Quote: | toni.lassila@gmail.com 작성:
mina_world wrote:
"The World Wide Wade" <waderameyxiii@comcast.remove13.net> wrote in message
In article <e9kds2$4is$1@news2.kornet.net>,
if A is a connected subset of X,
int(A), bd(A) is connected ?
To find a connected A such that int(A) is not connected, think
about subsets of R^2. (Try to find one so that both int(A) and
bd(A) are not connected.)
but i can't find the example with both int(A) and bd(A).
Take two disjoint infinite closed strips { (x,y) in R^2 : a <= x <= b,
y in R } and join them with a line segment in such a way that they
connect only in a boundary point.
yes, maybe this is a form of | |--| |
Two closed tangent disks with an interior point removed from one. |
|
|
| Back to top |
|
|
Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
|
| Posted: Wed Jul 19, 2006 4:12 pm Post subject:
Re: topology with connected.
|
|
|
In article <e9kds2$4is$1@news2.kornet.net>,
mina_world <mina_world@hanmail.net> wrote:
| Quote: | hello sir~
if A is a connected subset of X,
int(A), bd(A) is connected ?
|
The interior, no. Take two tangent discs on the plane with the usual
R^2 topology; the interior is two disjoint open balls.
The boundary, no. Take a closed interval in R with the usual topology;
the boundary is two isolated points.
| Quote: | ---------------------------------
um... i think..
if A = (1,2)U[2,3) in usual topology of R.
A is connected.
and
int(A) = (1,2)U(2,3) is not connected.
|
This is wrong. If you take A=(1,2) U [2,3), then you are taken
A=(1,3). So the interior is A itself.
| Quote: | bd(A) = {1,2,3} is not connected.
|
No, the boundary is {1,3}; 2 is an interior point. It is not
connected, however.
No on many counts, though two wrongs did make a right on the last
part.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin |
|
| Back to top |
|
|
mina_world
science forum Guru Wannabe
Joined: 20 Jul 2005
Posts: 186
|
| Posted: Thu Jul 20, 2006 5:06 am Post subject:
Re: topology with connected.
|
|
|
"William Elliot" <marsh@hevanet.remove.com> wrote in message
news:Pine.BSI.4.58.0607190359330.12173@vista.hevanet.com...
On Wed, 19 Jul 2006 mina_world@hanmail.net wrote:
| Quote: | toni.lassila@gmail.com ÀÛ¼º:
mina_world wrote:
"The World Wide Wade" <waderameyxiii@comcast.remove13.net> wrote in
message
In article <e9kds2$4is$1@news2.kornet.net>,
if A is a connected subset of X,
int(A), bd(A) is connected ?
To find a connected A such that int(A) is not connected, think
about subsets of R^2. (Try to find one so that both int(A) and
bd(A) are not connected.)
but i can't find the example with both int(A) and bd(A).
Take two disjoint infinite closed strips { (x,y) in R^2 : a <= x <= b,
y in R } and join them with a line segment in such a way that they
connect only in a boundary point.
yes, maybe this is a form of | |--| |
Two closed tangent disks with an interior point removed from one. |
sorry. since i had a weak english, i can't understand it.
http://board-2.blueweb.co.kr/user/math565/data/math/tangent.jpg
i need your advice one more. |
|
| Back to top |
|
|
William Elliot
science forum Guru
Joined: 24 Mar 2005
Posts: 1906
|
| Posted: Thu Jul 20, 2006 10:38 am Post subject:
Re: topology with connected.
|
|
|
On Thu, 20 Jul 2006, mina_world wrote:
Find connected S with int S disconnected, bd S disconnected
S = ({ (x,y) | (x - 1)^2 + y^2 <= 1 } \/
{ (x,y) | (x + 1)^2 + y^2 <= 1 }) - { (1,0) }
Similar example.
S = { (x,y) | 0 <= xy } - { (1,1) } |
|
| Back to top |
|
|
mina_world
science forum Guru Wannabe
Joined: 20 Jul 2005
Posts: 186
|
| Posted: Thu Jul 20, 2006 12:00 pm Post subject:
Re: topology with connected.
|
|
|
William Elliot 작성:
| Quote: | On Thu, 20 Jul 2006, mina_world wrote:
Find connected S with int S disconnected, bd S disconnected
Two closed tangent disks with an interior point removed from one.
sorry. since i had a weak english, i can't understand it.
http://board-2.blueweb.co.kr/user/math565/data/math/tangent.jpg
i need your advice one more.
S = ({ (x,y) | (x - 1)^2 + y^2 <= 1 } \/
{ (x,y) | (x + 1)^2 + y^2 <= 1 }) - { (1,0) }
Similar example.
S = { (x,y) | 0 <= xy } - { (1,1) }
|
yes, i understanded. thank you very much. |
|
| Back to top |
|
|
Google
|
|
| Back to top |
|
|
|
|
The time now is Wed Jul 06, 2011 11:12 am | All times are GMT
|
|
Copyright © 2004-2005 DeniX Solutions SRL
|
|
Other DeniX Solutions sites:
Electronics forum |
Medicine forum |
Unix/Linux blog |
Unix/Linux documentation |
Unix/Linux forums |
send newsletters
|
| |
|
Powered by phpBB © 2001, 2005 phpBB Group
|
|
FAQ
Search
Memberlist
Usergroups
Profile
Log in to check your private messages
Log in
Forum index
»
Science and Technology
»
Math
