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In a metric space $(X,d)$, $partial A=emptysetquadforall Asubseteq X$. Is $(X,d)$ a discrete metric space?
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I have a problem in my book, which asks me
Prove that, in a discrete metric space $(X,d)$, $partial
A=emptysetquadforall Asubseteq X$. Prove that, $(X,d)$ is a
discrete metric space.
Then, the author asks - Is the converse true? i.e. If in a metric space $(X,d)$, $partial A=emptysetquadforall Asubseteq X$, then is $(X,d)$ a discrete metric space?
Can anybody help me in this regard? Thanks for your assistance in advance.
N.B. Here $partial A$ denotes the set of all boundary points of $A$ i.e. set of all those points in $X$ which are not interior nor exterior point of $A$.
metric-spaces
asked Jul 18 at 18:03
Biswarup Saha
2318
add a comment |Â
up vote
-2
down vote
favorite
I have a problem in my book, which asks me
Prove that, in a discrete metric space $(X,d)$, $partial
A=emptysetquadforall Asubseteq X$. Prove that, $(X,d)$ is a
discrete metric space.
Then, the author asks - Is the converse true? i.e. If in a metric space $(X,d)$, $partial A=emptysetquadforall Asubseteq X$, then is $(X,d)$ a discrete metric space?
Can anybody help me in this regard? Thanks for your assistance in advance.
N.B. Here $partial A$ denotes the set of all boundary points of $A$ i.e. set of all those points in $X$ which are not interior nor exterior point of $A$.
metric-spaces
asked Jul 18 at 18:03
Biswarup Saha
2318
add a comment |Â
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
I have a problem in my book, which asks me
Prove that, in a discrete metric space $(X,d)$, $partial
A=emptysetquadforall Asubseteq X$. Prove that, $(X,d)$ is a
discrete metric space.
Then, the author asks - Is the converse true? i.e. If in a metric space $(X,d)$, $partial A=emptysetquadforall Asubseteq X$, then is $(X,d)$ a discrete metric space?
Can anybody help me in this regard? Thanks for your assistance in advance.
N.B. Here $partial A$ denotes the set of all boundary points of $A$ i.e. set of all those points in $X$ which are not interior nor exterior point of $A$.
metric-spaces
asked Jul 18 at 18:03
Biswarup Saha
2318
I have a problem in my book, which asks me
Prove that, in a discrete metric space $(X,d)$, $partial
A=emptysetquadforall Asubseteq X$. Prove that, $(X,d)$ is a
discrete metric space.
Then, the author asks - Is the converse true? i.e. If in a metric space $(X,d)$, $partial A=emptysetquadforall Asubseteq X$, then is $(X,d)$ a discrete metric space?
Can anybody help me in this regard? Thanks for your assistance in advance.
N.B. Here $partial A$ denotes the set of all boundary points of $A$ i.e. set of all those points in $X$ which are not interior nor exterior point of $A$.
metric-spaces
asked Jul 18 at 18:03
Biswarup Saha
2318
asked Jul 18 at 18:03
Biswarup Saha
2318
asked Jul 18 at 18:03
Biswarup Saha
2318
asked Jul 18 at 18:03
Biswarup Saha
2318
2318
add a comment |Â
add a comment |Â
1 Answer
1
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votes
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If $X$ is not discrete, then there's non-closed subset $A$ of $X$. Take $xinoverline Asetminus A$. Then $xinpartial A$ and therefore $partial Aneqemptyset$.
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
Jose Carlos Santos, If $(X,d)$ is not discrete metric space, then how to show, there will surely exist a non-closed subset?
– Biswarup Saha
Jul 18 at 18:12
How does the author define discrete metric space? It's not a tricky question. My answer is right or wrong depending on that.
– José Carlos Santos
Jul 18 at 18:14
@BiswarupSaha If all sets would be closed, all sets would be open too (being complements of closed sets), ergo $X$ has the discrete topology.
– Henno Brandsma
Jul 18 at 21:44
In my reading, a discrete metric space is a metric space whose topology is discrete. That would make this answer valid. On the other hand, if by discrete metric space one insists on e.g. $d(x, y) =1$ then it's clearly not true.
– Berci
Jul 18 at 21:55
@Berci Exactly what I thought.
– José Carlos Santos
Jul 18 at 22:11
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
If $X$ is not discrete, then there's non-closed subset $A$ of $X$. Take $xinoverline Asetminus A$. Then $xinpartial A$ and therefore $partial Aneqemptyset$.
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
Jose Carlos Santos, If $(X,d)$ is not discrete metric space, then how to show, there will surely exist a non-closed subset?
– Biswarup Saha
Jul 18 at 18:12
How does the author define discrete metric space? It's not a tricky question. My answer is right or wrong depending on that.
– José Carlos Santos
Jul 18 at 18:14
@BiswarupSaha If all sets would be closed, all sets would be open too (being complements of closed sets), ergo $X$ has the discrete topology.
– Henno Brandsma
Jul 18 at 21:44
In my reading, a discrete metric space is a metric space whose topology is discrete. That would make this answer valid. On the other hand, if by discrete metric space one insists on e.g. $d(x, y) =1$ then it's clearly not true.
– Berci
Jul 18 at 21:55
@Berci Exactly what I thought.
– José Carlos Santos
Jul 18 at 22:11
add a comment |Â
up vote
0
down vote
If $X$ is not discrete, then there's non-closed subset $A$ of $X$. Take $xinoverline Asetminus A$. Then $xinpartial A$ and therefore $partial Aneqemptyset$.
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
Jose Carlos Santos, If $(X,d)$ is not discrete metric space, then how to show, there will surely exist a non-closed subset?
– Biswarup Saha
Jul 18 at 18:12
How does the author define discrete metric space? It's not a tricky question. My answer is right or wrong depending on that.
– José Carlos Santos
Jul 18 at 18:14
@BiswarupSaha If all sets would be closed, all sets would be open too (being complements of closed sets), ergo $X$ has the discrete topology.
– Henno Brandsma
Jul 18 at 21:44
In my reading, a discrete metric space is a metric space whose topology is discrete. That would make this answer valid. On the other hand, if by discrete metric space one insists on e.g. $d(x, y) =1$ then it's clearly not true.
– Berci
Jul 18 at 21:55
@Berci Exactly what I thought.
– José Carlos Santos
Jul 18 at 22:11
add a comment |Â
up vote
0
down vote
up vote
0
down vote
If $X$ is not discrete, then there's non-closed subset $A$ of $X$. Take $xinoverline Asetminus A$. Then $xinpartial A$ and therefore $partial Aneqemptyset$.
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
If $X$ is not discrete, then there's non-closed subset $A$ of $X$. Take $xinoverline Asetminus A$. Then $xinpartial A$ and therefore $partial Aneqemptyset$.
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
answered Jul 18 at 18:07
José Carlos Santos
114k1698177
114k1698177
Jose Carlos Santos, If $(X,d)$ is not discrete metric space, then how to show, there will surely exist a non-closed subset?
– Biswarup Saha
Jul 18 at 18:12
How does the author define discrete metric space? It's not a tricky question. My answer is right or wrong depending on that.
– José Carlos Santos
Jul 18 at 18:14
@BiswarupSaha If all sets would be closed, all sets would be open too (being complements of closed sets), ergo $X$ has the discrete topology.
– Henno Brandsma
Jul 18 at 21:44
In my reading, a discrete metric space is a metric space whose topology is discrete. That would make this answer valid. On the other hand, if by discrete metric space one insists on e.g. $d(x, y) =1$ then it's clearly not true.
– Berci
Jul 18 at 21:55
@Berci Exactly what I thought.
– José Carlos Santos
Jul 18 at 22:11
add a comment |Â
Jose Carlos Santos, If $(X,d)$ is not discrete metric space, then how to show, there will surely exist a non-closed subset?
– Biswarup Saha
Jul 18 at 18:12
How does the author define discrete metric space? It's not a tricky question. My answer is right or wrong depending on that.
– José Carlos Santos
Jul 18 at 18:14
@BiswarupSaha If all sets would be closed, all sets would be open too (being complements of closed sets), ergo $X$ has the discrete topology.
– Henno Brandsma
Jul 18 at 21:44
In my reading, a discrete metric space is a metric space whose topology is discrete. That would make this answer valid. On the other hand, if by discrete metric space one insists on e.g. $d(x, y) =1$ then it's clearly not true.
– Berci
Jul 18 at 21:55
@Berci Exactly what I thought.
– José Carlos Santos
Jul 18 at 22:11
Jose Carlos Santos, If $(X,d)$ is not discrete metric space, then how to show, there will surely exist a non-closed subset?
– Biswarup Saha
Jul 18 at 18:12
Jose Carlos Santos, If $(X,d)$ is not discrete metric space, then how to show, there will surely exist a non-closed subset?
– Biswarup Saha
Jul 18 at 18:12
How does the author define discrete metric space? It's not a tricky question. My answer is right or wrong depending on that.
– José Carlos Santos
Jul 18 at 18:14
How does the author define discrete metric space? It's not a tricky question. My answer is right or wrong depending on that.
– José Carlos Santos
Jul 18 at 18:14
@BiswarupSaha If all sets would be closed, all sets would be open too (being complements of closed sets), ergo $X$ has the discrete topology.
– Henno Brandsma
Jul 18 at 21:44
@BiswarupSaha If all sets would be closed, all sets would be open too (being complements of closed sets), ergo $X$ has the discrete topology.
– Henno Brandsma
Jul 18 at 21:44
In my reading, a discrete metric space is a metric space whose topology is discrete. That would make this answer valid. On the other hand, if by discrete metric space one insists on e.g. $d(x, y) =1$ then it's clearly not true.
– Berci
Jul 18 at 21:55
In my reading, a discrete metric space is a metric space whose topology is discrete. That would make this answer valid. On the other hand, if by discrete metric space one insists on e.g. $d(x, y) =1$ then it's clearly not true.
– Berci
Jul 18 at 21:55
@Berci Exactly what I thought.
– José Carlos Santos
Jul 18 at 22:11
@Berci Exactly what I thought.
– José Carlos Santos
Jul 18 at 22:11
add a comment |Â
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