Topology, metric space and finite subsets
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Given $E$ is a totally bounded subset of a metric space $X$, I wish to show then for every $varepsilon > 0$, there exists a finite subset ${X_l, ldots ,X_n}$ of $E$ such that $E subset bigcup_{k = 1}^{n} B(x_k, varepsilon)$.
I know a metric space $X$ is compact if and only if every collection $F$ of closed sets in $X$ with the finite intersection property has a nonempty intersection but this only follows quite trivially from the definition of compactness. Any specific help is appreciated!
real-analysis general-topology metric-spaces
edited Jan 10 at 20:49
Viktor Glombik
772527
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
$endgroup$
|
show 2 more comments
$begingroup$
Given $E$ is a totally bounded subset of a metric space $X$, I wish to show then for every $varepsilon > 0$, there exists a finite subset ${X_l, ldots ,X_n}$ of $E$ such that $E subset bigcup_{k = 1}^{n} B(x_k, varepsilon)$.
I know a metric space $X$ is compact if and only if every collection $F$ of closed sets in $X$ with the finite intersection property has a nonempty intersection but this only follows quite trivially from the definition of compactness. Any specific help is appreciated!
real-analysis general-topology metric-spaces
edited Jan 10 at 20:49
Viktor Glombik
772527
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
$endgroup$
$begingroup$
Hint: the definition of totally bounded is that $E$ can be covered by finitely many balls of any radius.
$endgroup$
– probably_someone
Nov 15 '17 at 21:11
1
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What is your definition of totally bounded?
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– Alex S
Nov 15 '17 at 21:16
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I suppose the general definition is For every r > 0 there is a finite cover of X consisting of balls of radius r?
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– Homaniac
Nov 15 '17 at 21:23
1
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Then you're already done, it seems.
$endgroup$
– Henno Brandsma
Nov 15 '17 at 22:35
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@HennoBrandsma A plausible way of interpreting the question as something not completely trivial is that the definition of total boundedness might have allowed the centers of the balls to be anywhere in $X$ whereas the question wants centers in $E$.
$endgroup$
– Andreas Blass
Nov 15 '17 at 23:31
|
show 2 more comments
$begingroup$
Given $E$ is a totally bounded subset of a metric space $X$, I wish to show then for every $varepsilon > 0$, there exists a finite subset ${X_l, ldots ,X_n}$ of $E$ such that $E subset bigcup_{k = 1}^{n} B(x_k, varepsilon)$.
I know a metric space $X$ is compact if and only if every collection $F$ of closed sets in $X$ with the finite intersection property has a nonempty intersection but this only follows quite trivially from the definition of compactness. Any specific help is appreciated!
real-analysis general-topology metric-spaces
edited Jan 10 at 20:49
Viktor Glombik
772527
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
$endgroup$
Given $E$ is a totally bounded subset of a metric space $X$, I wish to show then for every $varepsilon > 0$, there exists a finite subset ${X_l, ldots ,X_n}$ of $E$ such that $E subset bigcup_{k = 1}^{n} B(x_k, varepsilon)$.
I know a metric space $X$ is compact if and only if every collection $F$ of closed sets in $X$ with the finite intersection property has a nonempty intersection but this only follows quite trivially from the definition of compactness. Any specific help is appreciated!
real-analysis general-topology metric-spaces
real-analysis general-topology metric-spaces
edited Jan 10 at 20:49
Viktor Glombik
772527
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
edited Jan 10 at 20:49
Viktor Glombik
772527
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
edited Jan 10 at 20:49
Viktor Glombik
772527
edited Jan 10 at 20:49
Viktor Glombik
772527
edited Jan 10 at 20:49
Viktor Glombik
772527
772527
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
asked Nov 15 '17 at 21:07
HomaniacHomaniac
586110
586110
$begingroup$
Hint: the definition of totally bounded is that $E$ can be covered by finitely many balls of any radius.
$endgroup$
– probably_someone
Nov 15 '17 at 21:11
1
$begingroup$
What is your definition of totally bounded?
$endgroup$
– Alex S
Nov 15 '17 at 21:16
$begingroup$
I suppose the general definition is For every r > 0 there is a finite cover of X consisting of balls of radius r?
$endgroup$
– Homaniac
Nov 15 '17 at 21:23
1
$begingroup$
Then you're already done, it seems.
$endgroup$
– Henno Brandsma
Nov 15 '17 at 22:35
$begingroup$
@HennoBrandsma A plausible way of interpreting the question as something not completely trivial is that the definition of total boundedness might have allowed the centers of the balls to be anywhere in $X$ whereas the question wants centers in $E$.
$endgroup$
– Andreas Blass
Nov 15 '17 at 23:31
|
show 2 more comments
$begingroup$
Hint: the definition of totally bounded is that $E$ can be covered by finitely many balls of any radius.
$endgroup$
– probably_someone
Nov 15 '17 at 21:11
1
$begingroup$
What is your definition of totally bounded?
$endgroup$
– Alex S
Nov 15 '17 at 21:16
$begingroup$
I suppose the general definition is For every r > 0 there is a finite cover of X consisting of balls of radius r?
$endgroup$
– Homaniac
Nov 15 '17 at 21:23
1
$begingroup$
Then you're already done, it seems.
$endgroup$
– Henno Brandsma
Nov 15 '17 at 22:35
$begingroup$
@HennoBrandsma A plausible way of interpreting the question as something not completely trivial is that the definition of total boundedness might have allowed the centers of the balls to be anywhere in $X$ whereas the question wants centers in $E$.
$endgroup$
– Andreas Blass
Nov 15 '17 at 23:31
$begingroup$
Hint: the definition of totally bounded is that $E$ can be covered by finitely many balls of any radius.
$endgroup$
– probably_someone
Nov 15 '17 at 21:11
$begingroup$
Hint: the definition of totally bounded is that $E$ can be covered by finitely many balls of any radius.
$endgroup$
– probably_someone
Nov 15 '17 at 21:11
1
1
$begingroup$
What is your definition of totally bounded?
$endgroup$
– Alex S
Nov 15 '17 at 21:16
$begingroup$
What is your definition of totally bounded?
$endgroup$
– Alex S
Nov 15 '17 at 21:16
$begingroup$
I suppose the general definition is For every r > 0 there is a finite cover of X consisting of balls of radius r?
$endgroup$
– Homaniac
Nov 15 '17 at 21:23
$begingroup$
I suppose the general definition is For every r > 0 there is a finite cover of X consisting of balls of radius r?
$endgroup$
– Homaniac
Nov 15 '17 at 21:23
1
1
$begingroup$
Then you're already done, it seems.
$endgroup$
– Henno Brandsma
Nov 15 '17 at 22:35
$begingroup$
Then you're already done, it seems.
$endgroup$
– Henno Brandsma
Nov 15 '17 at 22:35
$begingroup$
@HennoBrandsma A plausible way of interpreting the question as something not completely trivial is that the definition of total boundedness might have allowed the centers of the balls to be anywhere in $X$ whereas the question wants centers in $E$.
$endgroup$
– Andreas Blass
Nov 15 '17 at 23:31
$begingroup$
@HennoBrandsma A plausible way of interpreting the question as something not completely trivial is that the definition of total boundedness might have allowed the centers of the balls to be anywhere in $X$ whereas the question wants centers in $E$.
$endgroup$
– Andreas Blass
Nov 15 '17 at 23:31
|
show 2 more comments
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$begingroup$
Hint: the definition of totally bounded is that $E$ can be covered by finitely many balls of any radius.
$endgroup$
– probably_someone
Nov 15 '17 at 21:11
1
$begingroup$
What is your definition of totally bounded?
$endgroup$
– Alex S
Nov 15 '17 at 21:16
$begingroup$
I suppose the general definition is For every r > 0 there is a finite cover of X consisting of balls of radius r?
$endgroup$
– Homaniac
Nov 15 '17 at 21:23
1
$begingroup$
Then you're already done, it seems.
$endgroup$
– Henno Brandsma
Nov 15 '17 at 22:35
$begingroup$
@HennoBrandsma A plausible way of interpreting the question as something not completely trivial is that the definition of total boundedness might have allowed the centers of the balls to be anywhere in $X$ whereas the question wants centers in $E$.
$endgroup$
– Andreas Blass
Nov 15 '17 at 23:31