Can ANY compact set in $mathbb R$ (which is not closed and bounded interval and $varnothing$) be written as $[a,b] - bigcuplimits_n I_n$?
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Let K be a non-empty compact subset of $mathbb R$ . Prove that $K$ is of the form $[a,b]$ or of the form $$[a,b] - bigcuplimits_n I_n $$ where $I_n $ is a countable disjoint family of open intervals with end points in $K$.
Will this statement be true?
My Try : I think this statement is true. As $K$ will be contained in any closed interval $[a,b]$ where $a , b in K$. and $K$ will be a closed So $mathbb R - K$ is the union of countable union of disjoint open intervals . $mathbb R - K = bigcuplimits_n I_n$.$;$ So $K = [a,b] - bigcuplimits_n I_n $
Am I correct? Can anyone please mention where did I wrong?
real-analysis general-topology compactness
edited Sep 6 at 8:04
Davide Morgante
2,549723
asked Sep 6 at 7:27
cmi
951110
 |Â
show 4 more comments
up vote
2
down vote
favorite
Let K be a non-empty compact subset of $mathbb R$ . Prove that $K$ is of the form $[a,b]$ or of the form $$[a,b] - bigcuplimits_n I_n $$ where $I_n $ is a countable disjoint family of open intervals with end points in $K$.
Will this statement be true?
My Try : I think this statement is true. As $K$ will be contained in any closed interval $[a,b]$ where $a , b in K$. and $K$ will be a closed So $mathbb R - K$ is the union of countable union of disjoint open intervals . $mathbb R - K = bigcuplimits_n I_n$.$;$ So $K = [a,b] - bigcuplimits_n I_n $
Am I correct? Can anyone please mention where did I wrong?
real-analysis general-topology compactness
edited Sep 6 at 8:04
Davide Morgante
2,549723
asked Sep 6 at 7:27
cmi
951110
Why do you think you did something wrong? Your proof is correct.
– Kavi Rama Murthy
Sep 6 at 7:30
How about $varnothing$?
– Sobi
Sep 6 at 7:33
Actually some problems have been posted earlier where the compact set $K$ had to have non empty interior . @KaviRamaMurthy
– cmi
Sep 6 at 7:33
edited....……. @Sobi
– cmi
Sep 6 at 7:34
1
@WilliamElliot It is assumed that the endpoints of the open intervals are in $K$.
– Sobi
Sep 6 at 11:55
 |Â
show 4 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let K be a non-empty compact subset of $mathbb R$ . Prove that $K$ is of the form $[a,b]$ or of the form $$[a,b] - bigcuplimits_n I_n $$ where $I_n $ is a countable disjoint family of open intervals with end points in $K$.
Will this statement be true?
My Try : I think this statement is true. As $K$ will be contained in any closed interval $[a,b]$ where $a , b in K$. and $K$ will be a closed So $mathbb R - K$ is the union of countable union of disjoint open intervals . $mathbb R - K = bigcuplimits_n I_n$.$;$ So $K = [a,b] - bigcuplimits_n I_n $
Am I correct? Can anyone please mention where did I wrong?
real-analysis general-topology compactness
edited Sep 6 at 8:04
Davide Morgante
2,549723
asked Sep 6 at 7:27
cmi
951110
Let K be a non-empty compact subset of $mathbb R$ . Prove that $K$ is of the form $[a,b]$ or of the form $$[a,b] - bigcuplimits_n I_n $$ where $I_n $ is a countable disjoint family of open intervals with end points in $K$.
Will this statement be true?
My Try : I think this statement is true. As $K$ will be contained in any closed interval $[a,b]$ where $a , b in K$. and $K$ will be a closed So $mathbb R - K$ is the union of countable union of disjoint open intervals . $mathbb R - K = bigcuplimits_n I_n$.$;$ So $K = [a,b] - bigcuplimits_n I_n $
Am I correct? Can anyone please mention where did I wrong?
real-analysis general-topology compactness
real-analysis general-topology compactness
edited Sep 6 at 8:04
Davide Morgante
2,549723
asked Sep 6 at 7:27
cmi
951110
edited Sep 6 at 8:04
Davide Morgante
2,549723
asked Sep 6 at 7:27
cmi
951110
edited Sep 6 at 8:04
Davide Morgante
2,549723
edited Sep 6 at 8:04
Davide Morgante
2,549723
edited Sep 6 at 8:04
Davide Morgante
2,549723
2,549723
asked Sep 6 at 7:27
cmi
951110
asked Sep 6 at 7:27
cmi
951110
asked Sep 6 at 7:27
cmi
951110
951110
Why do you think you did something wrong? Your proof is correct.
– Kavi Rama Murthy
Sep 6 at 7:30
How about $varnothing$?
– Sobi
Sep 6 at 7:33
Actually some problems have been posted earlier where the compact set $K$ had to have non empty interior . @KaviRamaMurthy
– cmi
Sep 6 at 7:33
edited....……. @Sobi
– cmi
Sep 6 at 7:34
1
@WilliamElliot It is assumed that the endpoints of the open intervals are in $K$.
– Sobi
Sep 6 at 11:55
 |Â
show 4 more comments
Why do you think you did something wrong? Your proof is correct.
– Kavi Rama Murthy
Sep 6 at 7:30
How about $varnothing$?
– Sobi
Sep 6 at 7:33
Actually some problems have been posted earlier where the compact set $K$ had to have non empty interior . @KaviRamaMurthy
– cmi
Sep 6 at 7:33
edited....……. @Sobi
– cmi
Sep 6 at 7:34
1
@WilliamElliot It is assumed that the endpoints of the open intervals are in $K$.
– Sobi
Sep 6 at 11:55
Why do you think you did something wrong? Your proof is correct.
– Kavi Rama Murthy
Sep 6 at 7:30
Why do you think you did something wrong? Your proof is correct.
– Kavi Rama Murthy
Sep 6 at 7:30
How about $varnothing$?
– Sobi
Sep 6 at 7:33
How about $varnothing$?
– Sobi
Sep 6 at 7:33
Actually some problems have been posted earlier where the compact set $K$ had to have non empty interior . @KaviRamaMurthy
– cmi
Sep 6 at 7:33
Actually some problems have been posted earlier where the compact set $K$ had to have non empty interior . @KaviRamaMurthy
– cmi
Sep 6 at 7:33
edited....……. @Sobi
– cmi
Sep 6 at 7:34
edited....……. @Sobi
– cmi
Sep 6 at 7:34
1
1
@WilliamElliot It is assumed that the endpoints of the open intervals are in $K$.
– Sobi
Sep 6 at 11:55
@WilliamElliot It is assumed that the endpoints of the open intervals are in $K$.
– Sobi
Sep 6 at 11:55
 |Â
show 4 more comments
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Why do you think you did something wrong? Your proof is correct.
– Kavi Rama Murthy
Sep 6 at 7:30
How about $varnothing$?
– Sobi
Sep 6 at 7:33
Actually some problems have been posted earlier where the compact set $K$ had to have non empty interior . @KaviRamaMurthy
– cmi
Sep 6 at 7:33
edited....……. @Sobi
– cmi
Sep 6 at 7:34
1
@WilliamElliot It is assumed that the endpoints of the open intervals are in $K$.
– Sobi
Sep 6 at 11:55