If a collection of closed sets of arbitrary cardinality in a metric space has empty intersection, does some countable subcollection?
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In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?
Is this possible in a metric space?
general-topology metric-spaces
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
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add a comment |
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In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?
Is this possible in a metric space?
general-topology metric-spaces
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
$endgroup$
add a comment |
$begingroup$
In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?
Is this possible in a metric space?
general-topology metric-spaces
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
$endgroup$
In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?
Is this possible in a metric space?
general-topology metric-spaces
general-topology metric-spaces
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
asked Feb 9 at 21:54
Matt SamuelMatt Samuel
38.7k63769
38.7k63769
add a comment |
add a comment |
2 Answers
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Let $X$ be an uncountable set endowed with the discrete metric. Then the family $Xsetminusx,$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
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Of course. Thank you.
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– Matt Samuel
Feb 9 at 22:46
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The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.
Hence Santos' example was the standard example of a non-separable metric space.
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
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active
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active
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$begingroup$
Let $X$ be an uncountable set endowed with the discrete metric. Then the family $Xsetminusx,$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
$endgroup$
$begingroup$
Of course. Thank you.
$endgroup$
– Matt Samuel
Feb 9 at 22:46
add a comment |
$begingroup$
Let $X$ be an uncountable set endowed with the discrete metric. Then the family $Xsetminusx,$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
$endgroup$
$begingroup$
Of course. Thank you.
$endgroup$
– Matt Samuel
Feb 9 at 22:46
add a comment |
$begingroup$
Let $X$ be an uncountable set endowed with the discrete metric. Then the family $Xsetminusx,$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
$endgroup$
Let $X$ be an uncountable set endowed with the discrete metric. Then the family $Xsetminusx,$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
answered Feb 9 at 22:11
José Carlos SantosJosé Carlos Santos
165k22132235
165k22132235
$begingroup$
Of course. Thank you.
$endgroup$
– Matt Samuel
Feb 9 at 22:46
add a comment |
$begingroup$
Of course. Thank you.
$endgroup$
– Matt Samuel
Feb 9 at 22:46
$begingroup$
Of course. Thank you.
$endgroup$
– Matt Samuel
Feb 9 at 22:46
$begingroup$
Of course. Thank you.
$endgroup$
– Matt Samuel
Feb 9 at 22:46
add a comment |
$begingroup$
The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.
Hence Santos' example was the standard example of a non-separable metric space.
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
$endgroup$
add a comment |
$begingroup$
The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.
Hence Santos' example was the standard example of a non-separable metric space.
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
$endgroup$
add a comment |
$begingroup$
The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.
Hence Santos' example was the standard example of a non-separable metric space.
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
$endgroup$
The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.
Hence Santos' example was the standard example of a non-separable metric space.
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
answered Feb 9 at 23:25
Henno BrandsmaHenno Brandsma
111k348120
111k348120
add a comment |
add a comment |
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