Mixing
About clopen set in $mathbb R^n$
$begingroup$
Is there any clopen set in the metric space ($mathbb R^n$,$d$) (where $d$ is the usual metric)
besides $mathbb R^n$ & the empty set?
metric-spaces
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
$endgroup$
add a comment |
$begingroup$
Is there any clopen set in the metric space ($mathbb R^n$,$d$) (where $d$ is the usual metric)
besides $mathbb R^n$ & the empty set?
metric-spaces
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
$endgroup$
add a comment |
$begingroup$
Is there any clopen set in the metric space ($mathbb R^n$,$d$) (where $d$ is the usual metric)
besides $mathbb R^n$ & the empty set?
metric-spaces
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
$endgroup$
Is there any clopen set in the metric space ($mathbb R^n$,$d$) (where $d$ is the usual metric)
besides $mathbb R^n$ & the empty set?
metric-spaces
metric-spaces
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
asked Feb 1 at 14:42
Supriyo BanerjeeSupriyo Banerjee
1056
1056
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
No. If there were such a set $U$, then setting $V = mathbb{R}^n setminus U$, $V$ will also be clopen, so that $mathbb{R}^n = U sqcup V$ is a decomposition of $mathbb{R}^n$ as a union of nonempty disjoint open sets.
answered Feb 1 at 14:46
cspruncsprun
2,804211
$endgroup$
$begingroup$
How can I prove $U$ and $V$ are separated?
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:08
$begingroup$
What do you mean?
$endgroup$
– csprun
Feb 1 at 15:08
$begingroup$
If U and V are separated and U $union$ V =$mathbb R^n$ then $mathbb R^n$ is not connected
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:11
1
$begingroup$
We defined $V$ so that it has nothing in common with $U$. (Also, the LaTeX command for the 'union' symbol is cup !)
$endgroup$
– csprun
Feb 1 at 15:12
add a comment |
$begingroup$
The metric spaces $X$ for which the only clopen subsets are $emptyset$ and $X$ have a name: connected spaces. And, yes, $mathbb{R}^n$ is connected. Therefore, it has no other clopen subsets.
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
It's more general than metric spaces!
$endgroup$
– csprun
Feb 1 at 15:11
$begingroup$
I know, but I suspect that the OP is not aware of topological spaces.
$endgroup$
– José Carlos Santos
Feb 1 at 15:12
$begingroup$
That's fair. I figured you did but wanted to add it in case OP does know about them.
$endgroup$
– csprun
Feb 1 at 15:13
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No. If there were such a set $U$, then setting $V = mathbb{R}^n setminus U$, $V$ will also be clopen, so that $mathbb{R}^n = U sqcup V$ is a decomposition of $mathbb{R}^n$ as a union of nonempty disjoint open sets.
answered Feb 1 at 14:46
cspruncsprun
2,804211
$endgroup$
$begingroup$
How can I prove $U$ and $V$ are separated?
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:08
$begingroup$
What do you mean?
$endgroup$
– csprun
Feb 1 at 15:08
$begingroup$
If U and V are separated and U $union$ V =$mathbb R^n$ then $mathbb R^n$ is not connected
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:11
1
$begingroup$
We defined $V$ so that it has nothing in common with $U$. (Also, the LaTeX command for the 'union' symbol is cup !)
$endgroup$
– csprun
Feb 1 at 15:12
add a comment |
$begingroup$
No. If there were such a set $U$, then setting $V = mathbb{R}^n setminus U$, $V$ will also be clopen, so that $mathbb{R}^n = U sqcup V$ is a decomposition of $mathbb{R}^n$ as a union of nonempty disjoint open sets.
answered Feb 1 at 14:46
cspruncsprun
2,804211
$endgroup$
$begingroup$
How can I prove $U$ and $V$ are separated?
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:08
$begingroup$
What do you mean?
$endgroup$
– csprun
Feb 1 at 15:08
$begingroup$
If U and V are separated and U $union$ V =$mathbb R^n$ then $mathbb R^n$ is not connected
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:11
1
$begingroup$
We defined $V$ so that it has nothing in common with $U$. (Also, the LaTeX command for the 'union' symbol is cup !)
$endgroup$
– csprun
Feb 1 at 15:12
add a comment |
$begingroup$
No. If there were such a set $U$, then setting $V = mathbb{R}^n setminus U$, $V$ will also be clopen, so that $mathbb{R}^n = U sqcup V$ is a decomposition of $mathbb{R}^n$ as a union of nonempty disjoint open sets.
answered Feb 1 at 14:46
cspruncsprun
2,804211
$endgroup$
No. If there were such a set $U$, then setting $V = mathbb{R}^n setminus U$, $V$ will also be clopen, so that $mathbb{R}^n = U sqcup V$ is a decomposition of $mathbb{R}^n$ as a union of nonempty disjoint open sets.
answered Feb 1 at 14:46
cspruncsprun
2,804211
edited Feb 1 at 15:09
answered Feb 1 at 14:46
cspruncsprun
2,804211
answered Feb 1 at 14:46
cspruncsprun
2,804211
answered Feb 1 at 14:46
cspruncsprun
2,804211
2,804211
$begingroup$
How can I prove $U$ and $V$ are separated?
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:08
$begingroup$
What do you mean?
$endgroup$
– csprun
Feb 1 at 15:08
$begingroup$
If U and V are separated and U $union$ V =$mathbb R^n$ then $mathbb R^n$ is not connected
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:11
1
$begingroup$
We defined $V$ so that it has nothing in common with $U$. (Also, the LaTeX command for the 'union' symbol is cup !)
$endgroup$
– csprun
Feb 1 at 15:12
add a comment |
$begingroup$
How can I prove $U$ and $V$ are separated?
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:08
$begingroup$
What do you mean?
$endgroup$
– csprun
Feb 1 at 15:08
$begingroup$
If U and V are separated and U $union$ V =$mathbb R^n$ then $mathbb R^n$ is not connected
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:11
1
$begingroup$
We defined $V$ so that it has nothing in common with $U$. (Also, the LaTeX command for the 'union' symbol is cup !)
$endgroup$
– csprun
Feb 1 at 15:12
$begingroup$
How can I prove $U$ and $V$ are separated?
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:08
$begingroup$
How can I prove $U$ and $V$ are separated?
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:08
$begingroup$
What do you mean?
$endgroup$
– csprun
Feb 1 at 15:08
$begingroup$
What do you mean?
$endgroup$
– csprun
Feb 1 at 15:08
$begingroup$
If U and V are separated and U $union$ V =$mathbb R^n$ then $mathbb R^n$ is not connected
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:11
$begingroup$
If U and V are separated and U $union$ V =$mathbb R^n$ then $mathbb R^n$ is not connected
$endgroup$
– Supriyo Banerjee
Feb 1 at 15:11
1
1
$begingroup$
We defined $V$ so that it has nothing in common with $U$. (Also, the LaTeX command for the 'union' symbol is cup !)
$endgroup$
– csprun
Feb 1 at 15:12
$begingroup$
We defined $V$ so that it has nothing in common with $U$. (Also, the LaTeX command for the 'union' symbol is cup !)
$endgroup$
– csprun
Feb 1 at 15:12
add a comment |
$begingroup$
The metric spaces $X$ for which the only clopen subsets are $emptyset$ and $X$ have a name: connected spaces. And, yes, $mathbb{R}^n$ is connected. Therefore, it has no other clopen subsets.
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
It's more general than metric spaces!
$endgroup$
– csprun
Feb 1 at 15:11
$begingroup$
I know, but I suspect that the OP is not aware of topological spaces.
$endgroup$
– José Carlos Santos
Feb 1 at 15:12
$begingroup$
That's fair. I figured you did but wanted to add it in case OP does know about them.
$endgroup$
– csprun
Feb 1 at 15:13
add a comment |
$begingroup$
The metric spaces $X$ for which the only clopen subsets are $emptyset$ and $X$ have a name: connected spaces. And, yes, $mathbb{R}^n$ is connected. Therefore, it has no other clopen subsets.
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
It's more general than metric spaces!
$endgroup$
– csprun
Feb 1 at 15:11
$begingroup$
I know, but I suspect that the OP is not aware of topological spaces.
$endgroup$
– José Carlos Santos
Feb 1 at 15:12
$begingroup$
That's fair. I figured you did but wanted to add it in case OP does know about them.
$endgroup$
– csprun
Feb 1 at 15:13
add a comment |
$begingroup$
The metric spaces $X$ for which the only clopen subsets are $emptyset$ and $X$ have a name: connected spaces. And, yes, $mathbb{R}^n$ is connected. Therefore, it has no other clopen subsets.
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
The metric spaces $X$ for which the only clopen subsets are $emptyset$ and $X$ have a name: connected spaces. And, yes, $mathbb{R}^n$ is connected. Therefore, it has no other clopen subsets.
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
answered Feb 1 at 14:58
José Carlos SantosJosé Carlos Santos
174k23133242
174k23133242
$begingroup$
It's more general than metric spaces!
$endgroup$
– csprun
Feb 1 at 15:11
$begingroup$
I know, but I suspect that the OP is not aware of topological spaces.
$endgroup$
– José Carlos Santos
Feb 1 at 15:12
$begingroup$
That's fair. I figured you did but wanted to add it in case OP does know about them.
$endgroup$
– csprun
Feb 1 at 15:13
add a comment |
$begingroup$
It's more general than metric spaces!
$endgroup$
– csprun
Feb 1 at 15:11
$begingroup$
I know, but I suspect that the OP is not aware of topological spaces.
$endgroup$
– José Carlos Santos
Feb 1 at 15:12
$begingroup$
That's fair. I figured you did but wanted to add it in case OP does know about them.
$endgroup$
– csprun
Feb 1 at 15:13
$begingroup$
It's more general than metric spaces!
$endgroup$
– csprun
Feb 1 at 15:11
$begingroup$
It's more general than metric spaces!
$endgroup$
– csprun
Feb 1 at 15:11
$begingroup$
I know, but I suspect that the OP is not aware of topological spaces.
$endgroup$
– José Carlos Santos
Feb 1 at 15:12
$begingroup$
I know, but I suspect that the OP is not aware of topological spaces.
$endgroup$
– José Carlos Santos
Feb 1 at 15:12
$begingroup$
That's fair. I figured you did but wanted to add it in case OP does know about them.
$endgroup$
– csprun
Feb 1 at 15:13
$begingroup$
That's fair. I figured you did but wanted to add it in case OP does know about them.
$endgroup$
– csprun
Feb 1 at 15:13
add a comment |
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