Cardinality of compacts with positive $(>0)$ Lebesgue measure in $Bbb R^3$
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I need to prove that it's same as $[0,1]$ (continuum).
Let's say I have proved "fact" that closed balls with positive radius in $Bbb R^3$ have same cardinality as $[0,1]$.
Does it prove that compacts have cardinality no more than continuum? If so, what I should do in "no less than" part (or vice versa no less than -> no more than)?
real-analysis lebesgue-measure compactness
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
asked Nov 24 at 0:16
MrDdGANGER
82
add a comment |
up vote
1
down vote
favorite
I need to prove that it's same as $[0,1]$ (continuum).
Let's say I have proved "fact" that closed balls with positive radius in $Bbb R^3$ have same cardinality as $[0,1]$.
Does it prove that compacts have cardinality no more than continuum? If so, what I should do in "no less than" part (or vice versa no less than -> no more than)?
real-analysis lebesgue-measure compactness
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
asked Nov 24 at 0:16
MrDdGANGER
82
Associate a countable dense subset with each compact set.
– Kavi Rama Murthy
Nov 24 at 0:42
Sorry but i can't understand how this will help with cardinality of compacts. Maximum i know from my course is definition of dense "everywhere".
– MrDdGANGER
Nov 26 at 23:32
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I need to prove that it's same as $[0,1]$ (continuum).
Let's say I have proved "fact" that closed balls with positive radius in $Bbb R^3$ have same cardinality as $[0,1]$.
Does it prove that compacts have cardinality no more than continuum? If so, what I should do in "no less than" part (or vice versa no less than -> no more than)?
real-analysis lebesgue-measure compactness
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
asked Nov 24 at 0:16
MrDdGANGER
82
I need to prove that it's same as $[0,1]$ (continuum).
Let's say I have proved "fact" that closed balls with positive radius in $Bbb R^3$ have same cardinality as $[0,1]$.
Does it prove that compacts have cardinality no more than continuum? If so, what I should do in "no less than" part (or vice versa no less than -> no more than)?
real-analysis lebesgue-measure compactness
real-analysis lebesgue-measure compactness
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
asked Nov 24 at 0:16
MrDdGANGER
82
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
asked Nov 24 at 0:16
MrDdGANGER
82
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
edited Nov 24 at 0:21
Chinnapparaj R
4,6591825
4,6591825
asked Nov 24 at 0:16
MrDdGANGER
82
asked Nov 24 at 0:16
MrDdGANGER
82
asked Nov 24 at 0:16
MrDdGANGER
82
82
Associate a countable dense subset with each compact set.
– Kavi Rama Murthy
Nov 24 at 0:42
Sorry but i can't understand how this will help with cardinality of compacts. Maximum i know from my course is definition of dense "everywhere".
– MrDdGANGER
Nov 26 at 23:32
add a comment |
Associate a countable dense subset with each compact set.
– Kavi Rama Murthy
Nov 24 at 0:42
Sorry but i can't understand how this will help with cardinality of compacts. Maximum i know from my course is definition of dense "everywhere".
– MrDdGANGER
Nov 26 at 23:32
Associate a countable dense subset with each compact set.
– Kavi Rama Murthy
Nov 24 at 0:42
Associate a countable dense subset with each compact set.
– Kavi Rama Murthy
Nov 24 at 0:42
Sorry but i can't understand how this will help with cardinality of compacts. Maximum i know from my course is definition of dense "everywhere".
– MrDdGANGER
Nov 26 at 23:32
Sorry but i can't understand how this will help with cardinality of compacts. Maximum i know from my course is definition of dense "everywhere".
– MrDdGANGER
Nov 26 at 23:32
add a comment |
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Associate a countable dense subset with each compact set.
– Kavi Rama Murthy
Nov 24 at 0:42
Sorry but i can't understand how this will help with cardinality of compacts. Maximum i know from my course is definition of dense "everywhere".
– MrDdGANGER
Nov 26 at 23:32