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(Prove or Disprove) If $X$ is a metric space, then the Borel $sigma$-algebra ${cal B}_X$ contains every...
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I am tasked with determining whether the following statement is true or false:
If $X$ is a metric space, then the Borel $sigma$-algebra ${cal B}_X$ contains every countable subset of $X$.
I understand how the Borel $sigma$-algebra is constructed, but I have very little intuition into what sets it contains.
measure-theory
asked Jan 21 at 6:57
johnny133253johnny133253
321110
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add a comment |
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I am tasked with determining whether the following statement is true or false:
If $X$ is a metric space, then the Borel $sigma$-algebra ${cal B}_X$ contains every countable subset of $X$.
I understand how the Borel $sigma$-algebra is constructed, but I have very little intuition into what sets it contains.
measure-theory
asked Jan 21 at 6:57
johnny133253johnny133253
321110
$endgroup$
add a comment |
$begingroup$
I am tasked with determining whether the following statement is true or false:
If $X$ is a metric space, then the Borel $sigma$-algebra ${cal B}_X$ contains every countable subset of $X$.
I understand how the Borel $sigma$-algebra is constructed, but I have very little intuition into what sets it contains.
measure-theory
asked Jan 21 at 6:57
johnny133253johnny133253
321110
$endgroup$
I am tasked with determining whether the following statement is true or false:
If $X$ is a metric space, then the Borel $sigma$-algebra ${cal B}_X$ contains every countable subset of $X$.
I understand how the Borel $sigma$-algebra is constructed, but I have very little intuition into what sets it contains.
measure-theory
measure-theory
asked Jan 21 at 6:57
johnny133253johnny133253
321110
asked Jan 21 at 6:57
johnny133253johnny133253
321110
asked Jan 21 at 6:57
johnny133253johnny133253
321110
asked Jan 21 at 6:57
johnny133253johnny133253
321110
asked Jan 21 at 6:57
johnny133253johnny133253
321110
321110
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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First, note that it contains every singleton subset of $X$. For take any point $x in X$; then the intersection of all open balls around $x$ with radius $1/n$ lies in the Borel sigma-algebra, but this only contains $x$ since $X$ is a metric space.
Now take any countable subset $A$ of $X$. Since the Borel sigma-algebra contains all singletons ${x}$ for $xin A$ and it’s closed under countable unions, it contains $A$.
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
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Show that ${x}$ is closed for every $x$ in a metric space. It's also a countable intersection of open sets $B(x,frac1n)$ so in either case, singletons are in the Borel $sigma$-algebra. Countable sets are countable unions of those.
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
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active
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votes
$begingroup$
First, note that it contains every singleton subset of $X$. For take any point $x in X$; then the intersection of all open balls around $x$ with radius $1/n$ lies in the Borel sigma-algebra, but this only contains $x$ since $X$ is a metric space.
Now take any countable subset $A$ of $X$. Since the Borel sigma-algebra contains all singletons ${x}$ for $xin A$ and it’s closed under countable unions, it contains $A$.
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
$endgroup$
add a comment |
$begingroup$
First, note that it contains every singleton subset of $X$. For take any point $x in X$; then the intersection of all open balls around $x$ with radius $1/n$ lies in the Borel sigma-algebra, but this only contains $x$ since $X$ is a metric space.
Now take any countable subset $A$ of $X$. Since the Borel sigma-algebra contains all singletons ${x}$ for $xin A$ and it’s closed under countable unions, it contains $A$.
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
$endgroup$
add a comment |
$begingroup$
First, note that it contains every singleton subset of $X$. For take any point $x in X$; then the intersection of all open balls around $x$ with radius $1/n$ lies in the Borel sigma-algebra, but this only contains $x$ since $X$ is a metric space.
Now take any countable subset $A$ of $X$. Since the Borel sigma-algebra contains all singletons ${x}$ for $xin A$ and it’s closed under countable unions, it contains $A$.
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
$endgroup$
First, note that it contains every singleton subset of $X$. For take any point $x in X$; then the intersection of all open balls around $x$ with radius $1/n$ lies in the Borel sigma-algebra, but this only contains $x$ since $X$ is a metric space.
Now take any countable subset $A$ of $X$. Since the Borel sigma-algebra contains all singletons ${x}$ for $xin A$ and it’s closed under countable unions, it contains $A$.
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
answered Jan 21 at 7:02
SvanNSvanN
2,0661422
2,0661422
add a comment |
add a comment |
$begingroup$
Show that ${x}$ is closed for every $x$ in a metric space. It's also a countable intersection of open sets $B(x,frac1n)$ so in either case, singletons are in the Borel $sigma$-algebra. Countable sets are countable unions of those.
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
$endgroup$
add a comment |
$begingroup$
Show that ${x}$ is closed for every $x$ in a metric space. It's also a countable intersection of open sets $B(x,frac1n)$ so in either case, singletons are in the Borel $sigma$-algebra. Countable sets are countable unions of those.
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
$endgroup$
add a comment |
$begingroup$
Show that ${x}$ is closed for every $x$ in a metric space. It's also a countable intersection of open sets $B(x,frac1n)$ so in either case, singletons are in the Borel $sigma$-algebra. Countable sets are countable unions of those.
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
$endgroup$
Show that ${x}$ is closed for every $x$ in a metric space. It's also a countable intersection of open sets $B(x,frac1n)$ so in either case, singletons are in the Borel $sigma$-algebra. Countable sets are countable unions of those.
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
answered Jan 21 at 7:00
Henno BrandsmaHenno Brandsma
112k348120
112k348120
add a comment |
add a comment |
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