Examples for $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.
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Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.
I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.
Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.
And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.
general-topology
asked Jan 2 at 2:45
user549397user549397
1,7141618
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add a comment |
$begingroup$
Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.
I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.
Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.
And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.
general-topology
asked Jan 2 at 2:45
user549397user549397
1,7141618
$endgroup$
add a comment |
$begingroup$
Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.
I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.
Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.
And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.
general-topology
asked Jan 2 at 2:45
user549397user549397
1,7141618
$endgroup$
Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.
I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.
Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.
And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.
general-topology
general-topology
asked Jan 2 at 2:45
user549397user549397
1,7141618
asked Jan 2 at 2:45
user549397user549397
1,7141618
asked Jan 2 at 2:45
user549397user549397
1,7141618
asked Jan 2 at 2:45
user549397user549397
1,7141618
asked Jan 2 at 2:45
user549397user549397
1,7141618
1,7141618
add a comment |
add a comment |
1 Answer
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In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.
answered Jan 2 at 2:47
RandallRandall
10.8k11431
$endgroup$
$begingroup$
but $overline{A}bigcap B not= emptyset$
$endgroup$
– Joel Pereira
Jan 2 at 5:21
$begingroup$
@JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
$endgroup$
– Randall
Jan 2 at 5:22
$begingroup$
oh i read it as "A complement" as opposed to cl(A). my fault.
$endgroup$
– Joel Pereira
Jan 2 at 5:24
add a comment |
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1 Answer
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1 Answer
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$begingroup$
In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.
answered Jan 2 at 2:47
RandallRandall
10.8k11431
$endgroup$
$begingroup$
but $overline{A}bigcap B not= emptyset$
$endgroup$
– Joel Pereira
Jan 2 at 5:21
$begingroup$
@JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
$endgroup$
– Randall
Jan 2 at 5:22
$begingroup$
oh i read it as "A complement" as opposed to cl(A). my fault.
$endgroup$
– Joel Pereira
Jan 2 at 5:24
add a comment |
$begingroup$
In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.
answered Jan 2 at 2:47
RandallRandall
10.8k11431
$endgroup$
$begingroup$
but $overline{A}bigcap B not= emptyset$
$endgroup$
– Joel Pereira
Jan 2 at 5:21
$begingroup$
@JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
$endgroup$
– Randall
Jan 2 at 5:22
$begingroup$
oh i read it as "A complement" as opposed to cl(A). my fault.
$endgroup$
– Joel Pereira
Jan 2 at 5:24
add a comment |
$begingroup$
In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.
answered Jan 2 at 2:47
RandallRandall
10.8k11431
$endgroup$
In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.
answered Jan 2 at 2:47
RandallRandall
10.8k11431
edited Jan 2 at 2:51
answered Jan 2 at 2:47
RandallRandall
10.8k11431
answered Jan 2 at 2:47
RandallRandall
10.8k11431
answered Jan 2 at 2:47
RandallRandall
10.8k11431
10.8k11431
$begingroup$
but $overline{A}bigcap B not= emptyset$
$endgroup$
– Joel Pereira
Jan 2 at 5:21
$begingroup$
@JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
$endgroup$
– Randall
Jan 2 at 5:22
$begingroup$
oh i read it as "A complement" as opposed to cl(A). my fault.
$endgroup$
– Joel Pereira
Jan 2 at 5:24
add a comment |
$begingroup$
but $overline{A}bigcap B not= emptyset$
$endgroup$
– Joel Pereira
Jan 2 at 5:21
$begingroup$
@JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
$endgroup$
– Randall
Jan 2 at 5:22
$begingroup$
oh i read it as "A complement" as opposed to cl(A). my fault.
$endgroup$
– Joel Pereira
Jan 2 at 5:24
$begingroup$
but $overline{A}bigcap B not= emptyset$
$endgroup$
– Joel Pereira
Jan 2 at 5:21
$begingroup$
but $overline{A}bigcap B not= emptyset$
$endgroup$
– Joel Pereira
Jan 2 at 5:21
$begingroup$
@JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
$endgroup$
– Randall
Jan 2 at 5:22
$begingroup$
@JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
$endgroup$
– Randall
Jan 2 at 5:22
$begingroup$
oh i read it as "A complement" as opposed to cl(A). my fault.
$endgroup$
– Joel Pereira
Jan 2 at 5:24
$begingroup$
oh i read it as "A complement" as opposed to cl(A). my fault.
$endgroup$
– Joel Pereira
Jan 2 at 5:24
add a comment |
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