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topology with connected.
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mina_world
science forum Guru Wannabe


Joined: 20 Jul 2005
Posts: 186

PostPosted: Wed Jul 19, 2006 4:55 am    Post subject: topology with connected. Reply with 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 ?
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Wed Jul 19, 2006 6:09 am    Post subject: Re: topology with connected. Reply with quote

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
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The World Wide Wade
science forum Guru


Joined: 24 Mar 2005
Posts: 790

PostPosted: Wed Jul 19, 2006 6:26 am    Post subject: Re: topology with connected. Reply with quote

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.)
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mina_world
science forum Guru Wannabe


Joined: 20 Jul 2005
Posts: 186

PostPosted: Wed Jul 19, 2006 7:09 am    Post subject: Re: topology with connected. Reply with quote

"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).
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toni.lassila@gmail.com
science forum beginner


Joined: 18 May 2006
Posts: 2

PostPosted: Wed Jul 19, 2006 9:44 am    Post subject: Re: topology with connected. Reply with quote

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.
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mina_world
science forum Guru Wannabe


Joined: 20 Jul 2005
Posts: 186

PostPosted: Wed Jul 19, 2006 10:06 am    Post subject: Re: topology with connected. Reply with quote

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.
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Wed Jul 19, 2006 11:02 am    Post subject: Re: topology with connected. Reply with quote

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.
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Arturo Magidin
science forum Guru


Joined: 25 Mar 2005
Posts: 1838

PostPosted: Wed Jul 19, 2006 4:12 pm    Post subject: Re: topology with connected. Reply with quote

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.

Quote:
is this right ?

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
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mina_world
science forum Guru Wannabe


Joined: 20 Jul 2005
Posts: 186

PostPosted: Thu Jul 20, 2006 5:06 am    Post subject: Re: topology with connected. Reply with quote

"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 ۼ:
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.
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Thu Jul 20, 2006 10:38 am    Post subject: Re: topology with connected. Reply with quote

On Thu, 20 Jul 2006, mina_world wrote:

Find connected S with int S disconnected, bd S disconnected

Quote:
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) }
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mina_world
science forum Guru Wannabe


Joined: 20 Jul 2005
Posts: 186

PostPosted: Thu Jul 20, 2006 12:00 pm    Post subject: Re: topology with connected. Reply with quote

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.
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