Mixing
$bigcup_{xin K}B'(x,r)=left {xin X |d(x,K)leq rright }$ for $K$ compact
$begingroup$
Let $(X,d)$ be a metric space and let $K$ be a compact subset of $X$
Let $r>0$ be given
Prove, $bigcup_{xin K}B'(x,r)=left {xin X |d(x,K)leq rright }$ where $d(x,K)=inf_{yin K}d(x,y)$ (B' is closed ball)
Any hints ?
general-topology
edited Jan 21 at 10:34
Mariah
1,5561718
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
$endgroup$
add a comment |
$begingroup$
Let $(X,d)$ be a metric space and let $K$ be a compact subset of $X$
Let $r>0$ be given
Prove, $bigcup_{xin K}B'(x,r)=left {xin X |d(x,K)leq rright }$ where $d(x,K)=inf_{yin K}d(x,y)$ (B' is closed ball)
Any hints ?
general-topology
edited Jan 21 at 10:34
Mariah
1,5561718
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
$endgroup$
add a comment |
$begingroup$
Let $(X,d)$ be a metric space and let $K$ be a compact subset of $X$
Let $r>0$ be given
Prove, $bigcup_{xin K}B'(x,r)=left {xin X |d(x,K)leq rright }$ where $d(x,K)=inf_{yin K}d(x,y)$ (B' is closed ball)
Any hints ?
general-topology
edited Jan 21 at 10:34
Mariah
1,5561718
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
$endgroup$
Let $(X,d)$ be a metric space and let $K$ be a compact subset of $X$
Let $r>0$ be given
Prove, $bigcup_{xin K}B'(x,r)=left {xin X |d(x,K)leq rright }$ where $d(x,K)=inf_{yin K}d(x,y)$ (B' is closed ball)
Any hints ?
general-topology
general-topology
edited Jan 21 at 10:34
Mariah
1,5561718
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
edited Jan 21 at 10:34
Mariah
1,5561718
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
edited Jan 21 at 10:34
Mariah
1,5561718
edited Jan 21 at 10:34
Mariah
1,5561718
edited Jan 21 at 10:34
Mariah
1,5561718
1,5561718
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
asked Jan 20 at 22:53
Pedro AlvarèsPedro Alvarès
636
636
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hints:
$1). $ If $yin bigcup_{xin K}B'(x,r),$ then $yin B'(x,r)$ for some ball $B'$. Then, $d(x,y)le r.$
$2). $ If $yin left {xin X |d(x,K)leq rright }, $ then fix this $y$ and note that the function $d(y,cdot ):Xto mathbb R$ is continuous, so it attains its minimum on the compact set $K$.
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
$endgroup$
$begingroup$
I'm done with 1) , I found my way with the first inclusion,but I'm stuck with 2)
$endgroup$
– Pedro Alvarès
Jan 20 at 23:47
$begingroup$
The key phrase is: " it attains its minimum on the compact set K."
$endgroup$
– Matematleta
Jan 20 at 23:51
$begingroup$
You mean inf(d(y,K))=<d(y,K) ?
$endgroup$
– Pedro Alvarès
Jan 21 at 0:00
$begingroup$
I mean the inf is reached. So now that means that there is a ___ in ___ such that ___.
$endgroup$
– Matematleta
Jan 21 at 0:01
$begingroup$
There is a sequence in K that converges ??
$endgroup$
– Pedro Alvarès
Jan 21 at 0:08
|
show 4 more comments
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hints:
$1). $ If $yin bigcup_{xin K}B'(x,r),$ then $yin B'(x,r)$ for some ball $B'$. Then, $d(x,y)le r.$
$2). $ If $yin left {xin X |d(x,K)leq rright }, $ then fix this $y$ and note that the function $d(y,cdot ):Xto mathbb R$ is continuous, so it attains its minimum on the compact set $K$.
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
$endgroup$
$begingroup$
I'm done with 1) , I found my way with the first inclusion,but I'm stuck with 2)
$endgroup$
– Pedro Alvarès
Jan 20 at 23:47
$begingroup$
The key phrase is: " it attains its minimum on the compact set K."
$endgroup$
– Matematleta
Jan 20 at 23:51
$begingroup$
You mean inf(d(y,K))=<d(y,K) ?
$endgroup$
– Pedro Alvarès
Jan 21 at 0:00
$begingroup$
I mean the inf is reached. So now that means that there is a ___ in ___ such that ___.
$endgroup$
– Matematleta
Jan 21 at 0:01
$begingroup$
There is a sequence in K that converges ??
$endgroup$
– Pedro Alvarès
Jan 21 at 0:08
|
show 4 more comments
$begingroup$
Hints:
$1). $ If $yin bigcup_{xin K}B'(x,r),$ then $yin B'(x,r)$ for some ball $B'$. Then, $d(x,y)le r.$
$2). $ If $yin left {xin X |d(x,K)leq rright }, $ then fix this $y$ and note that the function $d(y,cdot ):Xto mathbb R$ is continuous, so it attains its minimum on the compact set $K$.
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
$endgroup$
$begingroup$
I'm done with 1) , I found my way with the first inclusion,but I'm stuck with 2)
$endgroup$
– Pedro Alvarès
Jan 20 at 23:47
$begingroup$
The key phrase is: " it attains its minimum on the compact set K."
$endgroup$
– Matematleta
Jan 20 at 23:51
$begingroup$
You mean inf(d(y,K))=<d(y,K) ?
$endgroup$
– Pedro Alvarès
Jan 21 at 0:00
$begingroup$
I mean the inf is reached. So now that means that there is a ___ in ___ such that ___.
$endgroup$
– Matematleta
Jan 21 at 0:01
$begingroup$
There is a sequence in K that converges ??
$endgroup$
– Pedro Alvarès
Jan 21 at 0:08
|
show 4 more comments
$begingroup$
Hints:
$1). $ If $yin bigcup_{xin K}B'(x,r),$ then $yin B'(x,r)$ for some ball $B'$. Then, $d(x,y)le r.$
$2). $ If $yin left {xin X |d(x,K)leq rright }, $ then fix this $y$ and note that the function $d(y,cdot ):Xto mathbb R$ is continuous, so it attains its minimum on the compact set $K$.
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
$endgroup$
Hints:
$1). $ If $yin bigcup_{xin K}B'(x,r),$ then $yin B'(x,r)$ for some ball $B'$. Then, $d(x,y)le r.$
$2). $ If $yin left {xin X |d(x,K)leq rright }, $ then fix this $y$ and note that the function $d(y,cdot ):Xto mathbb R$ is continuous, so it attains its minimum on the compact set $K$.
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
answered Jan 20 at 23:36
MatematletaMatematleta
11.5k2920
11.5k2920
$begingroup$
I'm done with 1) , I found my way with the first inclusion,but I'm stuck with 2)
$endgroup$
– Pedro Alvarès
Jan 20 at 23:47
$begingroup$
The key phrase is: " it attains its minimum on the compact set K."
$endgroup$
– Matematleta
Jan 20 at 23:51
$begingroup$
You mean inf(d(y,K))=<d(y,K) ?
$endgroup$
– Pedro Alvarès
Jan 21 at 0:00
$begingroup$
I mean the inf is reached. So now that means that there is a ___ in ___ such that ___.
$endgroup$
– Matematleta
Jan 21 at 0:01
$begingroup$
There is a sequence in K that converges ??
$endgroup$
– Pedro Alvarès
Jan 21 at 0:08
|
show 4 more comments
$begingroup$
I'm done with 1) , I found my way with the first inclusion,but I'm stuck with 2)
$endgroup$
– Pedro Alvarès
Jan 20 at 23:47
$begingroup$
The key phrase is: " it attains its minimum on the compact set K."
$endgroup$
– Matematleta
Jan 20 at 23:51
$begingroup$
You mean inf(d(y,K))=<d(y,K) ?
$endgroup$
– Pedro Alvarès
Jan 21 at 0:00
$begingroup$
I mean the inf is reached. So now that means that there is a ___ in ___ such that ___.
$endgroup$
– Matematleta
Jan 21 at 0:01
$begingroup$
There is a sequence in K that converges ??
$endgroup$
– Pedro Alvarès
Jan 21 at 0:08
$begingroup$
I'm done with 1) , I found my way with the first inclusion,but I'm stuck with 2)
$endgroup$
– Pedro Alvarès
Jan 20 at 23:47
$begingroup$
I'm done with 1) , I found my way with the first inclusion,but I'm stuck with 2)
$endgroup$
– Pedro Alvarès
Jan 20 at 23:47
$begingroup$
The key phrase is: " it attains its minimum on the compact set K."
$endgroup$
– Matematleta
Jan 20 at 23:51
$begingroup$
The key phrase is: " it attains its minimum on the compact set K."
$endgroup$
– Matematleta
Jan 20 at 23:51
$begingroup$
You mean inf(d(y,K))=<d(y,K) ?
$endgroup$
– Pedro Alvarès
Jan 21 at 0:00
$begingroup$
You mean inf(d(y,K))=<d(y,K) ?
$endgroup$
– Pedro Alvarès
Jan 21 at 0:00
$begingroup$
I mean the inf is reached. So now that means that there is a ___ in ___ such that ___.
$endgroup$
– Matematleta
Jan 21 at 0:01
$begingroup$
I mean the inf is reached. So now that means that there is a ___ in ___ such that ___.
$endgroup$
– Matematleta
Jan 21 at 0:01
$begingroup$
There is a sequence in K that converges ??
$endgroup$
– Pedro Alvarès
Jan 21 at 0:08
$begingroup$
There is a sequence in K that converges ??
$endgroup$
– Pedro Alvarès
Jan 21 at 0:08
|
show 4 more comments
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