Locally Complete Metric Space
$begingroup$
It is undoubtedly the concept of "Complete Metric Space" is well known for all the users.
The Locally Complete Metric Space is a metric space where each point has a neighbourhood (which is closed) is a complete metric space.
It is so obvious that every complete metric space is locally complete metric space.
I am looking for an example of locally complete metric space that is not complete metric space.
general-topology functional-analysis metric-spaces complete-spaces
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
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add a comment |
$begingroup$
It is undoubtedly the concept of "Complete Metric Space" is well known for all the users.
The Locally Complete Metric Space is a metric space where each point has a neighbourhood (which is closed) is a complete metric space.
It is so obvious that every complete metric space is locally complete metric space.
I am looking for an example of locally complete metric space that is not complete metric space.
general-topology functional-analysis metric-spaces complete-spaces
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
$endgroup$
add a comment |
$begingroup$
It is undoubtedly the concept of "Complete Metric Space" is well known for all the users.
The Locally Complete Metric Space is a metric space where each point has a neighbourhood (which is closed) is a complete metric space.
It is so obvious that every complete metric space is locally complete metric space.
I am looking for an example of locally complete metric space that is not complete metric space.
general-topology functional-analysis metric-spaces complete-spaces
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
$endgroup$
It is undoubtedly the concept of "Complete Metric Space" is well known for all the users.
The Locally Complete Metric Space is a metric space where each point has a neighbourhood (which is closed) is a complete metric space.
It is so obvious that every complete metric space is locally complete metric space.
I am looking for an example of locally complete metric space that is not complete metric space.
general-topology functional-analysis metric-spaces complete-spaces
general-topology functional-analysis metric-spaces complete-spaces
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
asked Nov 26 '18 at 22:06
Neil hawkingNeil hawking
49619
49619
add a comment |
add a comment |
1 Answer
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$begingroup$
Example 1: $lbrace dfrac1n: n in Bbb Nrbrace$ with the usual metric.
Example 2: Any open but non-closed subset $U$ of a complete metric space. Like open intervals of finite length in $Bbb R$.
How to bring these examples together? I think the following characterisation is valid: Let $A$ be a subset of a complete metric space. Then $A$ with the restricted metric is locally complete but not complete if and only if:
$A$ is not closed
$cl(cl(A) setminus A) cap A = emptyset$, or equivalently, no limit point of $cl(A)setminus A$ lies in $A$, or equivalently, for every $ain A$, $inf lbrace d(a, x): xin cl(A)setminus Arbrace > 0$.
Since every metric space embeds into its completion, this gives a full (well, up to describing completions and checking these properties ...) characterisation of the spaces you ask for.
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
Example 1: $lbrace dfrac1n: n in Bbb Nrbrace$ with the usual metric.
Example 2: Any open but non-closed subset $U$ of a complete metric space. Like open intervals of finite length in $Bbb R$.
How to bring these examples together? I think the following characterisation is valid: Let $A$ be a subset of a complete metric space. Then $A$ with the restricted metric is locally complete but not complete if and only if:
$A$ is not closed
$cl(cl(A) setminus A) cap A = emptyset$, or equivalently, no limit point of $cl(A)setminus A$ lies in $A$, or equivalently, for every $ain A$, $inf lbrace d(a, x): xin cl(A)setminus Arbrace > 0$.
Since every metric space embeds into its completion, this gives a full (well, up to describing completions and checking these properties ...) characterisation of the spaces you ask for.
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
$endgroup$
add a comment |
$begingroup$
Example 1: $lbrace dfrac1n: n in Bbb Nrbrace$ with the usual metric.
Example 2: Any open but non-closed subset $U$ of a complete metric space. Like open intervals of finite length in $Bbb R$.
How to bring these examples together? I think the following characterisation is valid: Let $A$ be a subset of a complete metric space. Then $A$ with the restricted metric is locally complete but not complete if and only if:
$A$ is not closed
$cl(cl(A) setminus A) cap A = emptyset$, or equivalently, no limit point of $cl(A)setminus A$ lies in $A$, or equivalently, for every $ain A$, $inf lbrace d(a, x): xin cl(A)setminus Arbrace > 0$.
Since every metric space embeds into its completion, this gives a full (well, up to describing completions and checking these properties ...) characterisation of the spaces you ask for.
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
$endgroup$
add a comment |
$begingroup$
Example 1: $lbrace dfrac1n: n in Bbb Nrbrace$ with the usual metric.
Example 2: Any open but non-closed subset $U$ of a complete metric space. Like open intervals of finite length in $Bbb R$.
How to bring these examples together? I think the following characterisation is valid: Let $A$ be a subset of a complete metric space. Then $A$ with the restricted metric is locally complete but not complete if and only if:
$A$ is not closed
$cl(cl(A) setminus A) cap A = emptyset$, or equivalently, no limit point of $cl(A)setminus A$ lies in $A$, or equivalently, for every $ain A$, $inf lbrace d(a, x): xin cl(A)setminus Arbrace > 0$.
Since every metric space embeds into its completion, this gives a full (well, up to describing completions and checking these properties ...) characterisation of the spaces you ask for.
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
$endgroup$
Example 1: $lbrace dfrac1n: n in Bbb Nrbrace$ with the usual metric.
Example 2: Any open but non-closed subset $U$ of a complete metric space. Like open intervals of finite length in $Bbb R$.
How to bring these examples together? I think the following characterisation is valid: Let $A$ be a subset of a complete metric space. Then $A$ with the restricted metric is locally complete but not complete if and only if:
$A$ is not closed
$cl(cl(A) setminus A) cap A = emptyset$, or equivalently, no limit point of $cl(A)setminus A$ lies in $A$, or equivalently, for every $ain A$, $inf lbrace d(a, x): xin cl(A)setminus Arbrace > 0$.
Since every metric space embeds into its completion, this gives a full (well, up to describing completions and checking these properties ...) characterisation of the spaces you ask for.
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
edited Nov 27 '18 at 6:25
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
answered Nov 26 '18 at 22:21
Torsten SchoenebergTorsten Schoeneberg
3,9562833
3,9562833
add a comment |
add a comment |
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