Finite subcover with special property
$begingroup$
Let $M$ compact and $mathcal{U}$ an open covering of $M$ that satisfies: each $p in M$ is contained in at least two members of $mathcal{U}$. Show that $mathcal{U}$ has a finite subcovering with the same property.
Writing $mathcal{U} = {U_{i}}$, since $M$ is compact, we have
$$M subset U_{alpha_{1}} cup cdots cup U_{alpha_{k}}.$$
How to ensure that taken $p in M$, $p in U_{alpha_{n}}cap U_{alpha_{m}}$? Intuitively, could it not be $p in U_{alpha_{1}}cap U_{alpha_{k+1}}$? I really have no idea how to prove it. Can someone give me a hint?
general-topology metric-spaces compactness
$endgroup$
add a comment |
$begingroup$
Let $M$ compact and $mathcal{U}$ an open covering of $M$ that satisfies: each $p in M$ is contained in at least two members of $mathcal{U}$. Show that $mathcal{U}$ has a finite subcovering with the same property.
Writing $mathcal{U} = {U_{i}}$, since $M$ is compact, we have
$$M subset U_{alpha_{1}} cup cdots cup U_{alpha_{k}}.$$
How to ensure that taken $p in M$, $p in U_{alpha_{n}}cap U_{alpha_{m}}$? Intuitively, could it not be $p in U_{alpha_{1}}cap U_{alpha_{k+1}}$? I really have no idea how to prove it. Can someone give me a hint?
general-topology metric-spaces compactness
$endgroup$
add a comment |
$begingroup$
Let $M$ compact and $mathcal{U}$ an open covering of $M$ that satisfies: each $p in M$ is contained in at least two members of $mathcal{U}$. Show that $mathcal{U}$ has a finite subcovering with the same property.
Writing $mathcal{U} = {U_{i}}$, since $M$ is compact, we have
$$M subset U_{alpha_{1}} cup cdots cup U_{alpha_{k}}.$$
How to ensure that taken $p in M$, $p in U_{alpha_{n}}cap U_{alpha_{m}}$? Intuitively, could it not be $p in U_{alpha_{1}}cap U_{alpha_{k+1}}$? I really have no idea how to prove it. Can someone give me a hint?
general-topology metric-spaces compactness
$endgroup$
Let $M$ compact and $mathcal{U}$ an open covering of $M$ that satisfies: each $p in M$ is contained in at least two members of $mathcal{U}$. Show that $mathcal{U}$ has a finite subcovering with the same property.
Writing $mathcal{U} = {U_{i}}$, since $M$ is compact, we have
$$M subset U_{alpha_{1}} cup cdots cup U_{alpha_{k}}.$$
How to ensure that taken $p in M$, $p in U_{alpha_{n}}cap U_{alpha_{m}}$? Intuitively, could it not be $p in U_{alpha_{1}}cap U_{alpha_{k+1}}$? I really have no idea how to prove it. Can someone give me a hint?
general-topology metric-spaces compactness
general-topology metric-spaces compactness
edited Dec 26 '18 at 22:28
Kenny Wong
18.6k21439
18.6k21439
asked Dec 26 '18 at 22:20
Lucas CorrêaLucas Corrêa
1,6181321
1,6181321
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1 Answer
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$begingroup$
Hint: Let $mathcal U = { U_i : iin I}$ be the original open covering of $M$.
Can you show that $mathcal V = { U_i cap U_j : i, j in I, i neq j }$ is also an open covering of $M$?
Then use the compactness of $M$ to find a finite refinement of $mathcal V$...
$endgroup$
$begingroup$
Since every $p in M$ is contained in at least two members of $mathcal{U}$, each $p$ is contained in $U_{i} cap U_{j}$ for $i neq j$, so $displaystyle bigcup_{i,jmid i neq j} U_{i} cap U_{j}$ cover $M$. Since $M$ is compact, we can extract a finite subcovering that satisfies the property.
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:34
$begingroup$
Is this a good idea?
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:35
1
$begingroup$
@LucasCorrêa Yes, that's the right idea.
$endgroup$
– Kenny Wong
Dec 26 '18 at 22:35
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Let $mathcal U = { U_i : iin I}$ be the original open covering of $M$.
Can you show that $mathcal V = { U_i cap U_j : i, j in I, i neq j }$ is also an open covering of $M$?
Then use the compactness of $M$ to find a finite refinement of $mathcal V$...
$endgroup$
$begingroup$
Since every $p in M$ is contained in at least two members of $mathcal{U}$, each $p$ is contained in $U_{i} cap U_{j}$ for $i neq j$, so $displaystyle bigcup_{i,jmid i neq j} U_{i} cap U_{j}$ cover $M$. Since $M$ is compact, we can extract a finite subcovering that satisfies the property.
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:34
$begingroup$
Is this a good idea?
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:35
1
$begingroup$
@LucasCorrêa Yes, that's the right idea.
$endgroup$
– Kenny Wong
Dec 26 '18 at 22:35
add a comment |
$begingroup$
Hint: Let $mathcal U = { U_i : iin I}$ be the original open covering of $M$.
Can you show that $mathcal V = { U_i cap U_j : i, j in I, i neq j }$ is also an open covering of $M$?
Then use the compactness of $M$ to find a finite refinement of $mathcal V$...
$endgroup$
$begingroup$
Since every $p in M$ is contained in at least two members of $mathcal{U}$, each $p$ is contained in $U_{i} cap U_{j}$ for $i neq j$, so $displaystyle bigcup_{i,jmid i neq j} U_{i} cap U_{j}$ cover $M$. Since $M$ is compact, we can extract a finite subcovering that satisfies the property.
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:34
$begingroup$
Is this a good idea?
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:35
1
$begingroup$
@LucasCorrêa Yes, that's the right idea.
$endgroup$
– Kenny Wong
Dec 26 '18 at 22:35
add a comment |
$begingroup$
Hint: Let $mathcal U = { U_i : iin I}$ be the original open covering of $M$.
Can you show that $mathcal V = { U_i cap U_j : i, j in I, i neq j }$ is also an open covering of $M$?
Then use the compactness of $M$ to find a finite refinement of $mathcal V$...
$endgroup$
Hint: Let $mathcal U = { U_i : iin I}$ be the original open covering of $M$.
Can you show that $mathcal V = { U_i cap U_j : i, j in I, i neq j }$ is also an open covering of $M$?
Then use the compactness of $M$ to find a finite refinement of $mathcal V$...
answered Dec 26 '18 at 22:26
Kenny WongKenny Wong
18.6k21439
18.6k21439
$begingroup$
Since every $p in M$ is contained in at least two members of $mathcal{U}$, each $p$ is contained in $U_{i} cap U_{j}$ for $i neq j$, so $displaystyle bigcup_{i,jmid i neq j} U_{i} cap U_{j}$ cover $M$. Since $M$ is compact, we can extract a finite subcovering that satisfies the property.
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:34
$begingroup$
Is this a good idea?
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:35
1
$begingroup$
@LucasCorrêa Yes, that's the right idea.
$endgroup$
– Kenny Wong
Dec 26 '18 at 22:35
add a comment |
$begingroup$
Since every $p in M$ is contained in at least two members of $mathcal{U}$, each $p$ is contained in $U_{i} cap U_{j}$ for $i neq j$, so $displaystyle bigcup_{i,jmid i neq j} U_{i} cap U_{j}$ cover $M$. Since $M$ is compact, we can extract a finite subcovering that satisfies the property.
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:34
$begingroup$
Is this a good idea?
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:35
1
$begingroup$
@LucasCorrêa Yes, that's the right idea.
$endgroup$
– Kenny Wong
Dec 26 '18 at 22:35
$begingroup$
Since every $p in M$ is contained in at least two members of $mathcal{U}$, each $p$ is contained in $U_{i} cap U_{j}$ for $i neq j$, so $displaystyle bigcup_{i,jmid i neq j} U_{i} cap U_{j}$ cover $M$. Since $M$ is compact, we can extract a finite subcovering that satisfies the property.
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:34
$begingroup$
Since every $p in M$ is contained in at least two members of $mathcal{U}$, each $p$ is contained in $U_{i} cap U_{j}$ for $i neq j$, so $displaystyle bigcup_{i,jmid i neq j} U_{i} cap U_{j}$ cover $M$. Since $M$ is compact, we can extract a finite subcovering that satisfies the property.
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:34
$begingroup$
Is this a good idea?
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:35
$begingroup$
Is this a good idea?
$endgroup$
– Lucas Corrêa
Dec 26 '18 at 22:35
1
1
$begingroup$
@LucasCorrêa Yes, that's the right idea.
$endgroup$
– Kenny Wong
Dec 26 '18 at 22:35
$begingroup$
@LucasCorrêa Yes, that's the right idea.
$endgroup$
– Kenny Wong
Dec 26 '18 at 22:35
add a comment |
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