Prove that we can write any subset of Real numbers as a Union of closed subsets, or intersect of open...
$begingroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
$endgroup$
add a comment |
$begingroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
$endgroup$
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
add a comment |
$begingroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
$endgroup$
I've attempted to write an arbitrary subset of real numbers as the union of it's single points.
Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite)
also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open.
for example can we write [0,1) as the union of the sets
[0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?
general-topology
general-topology
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
edited Dec 18 '18 at 15:38
Arman_jr
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
asked Dec 18 '18 at 15:03
Arman_jrArman_jr
235
235
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
add a comment |
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09
add a comment |
1 Answer
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$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
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add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
$endgroup$
add a comment |
$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
$endgroup$
add a comment |
$begingroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
$endgroup$
Hint
$A= bigcup_{xin A} {x}$
Also
$A= bigcap_{xin A^c} mathbb{R}-{x}$
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
answered Dec 18 '18 at 15:13
Rakesh BhattRakesh Bhatt
967214
967214
add a comment |
add a comment |
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$begingroup$
The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements...
$endgroup$
– Ned
Dec 18 '18 at 15:09