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Compact set

  • Thread starter Mathsgirl
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  • #1
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Homework Statement



R>0, let K be a closed subset of C such that K [tex]\subset[/tex] BR(0) (so K is compact). Show that there exists 0 < r < R such that K[tex]\subset[/tex] Br(0).

Homework Equations





The Attempt at a Solution



Can I write BR(0) = {x[tex]\in[/tex]C : d(x,0) [tex]\leq[/tex]R}?
I know that a compact set is closed and bounded.
Is it something to do with us using [tex]\subset[/tex] and not [tex]\subseteq[/tex]?
As if it was [tex]\subseteq[/tex] then maybe K = BR(0) then there wouldn't be an r. But as K is strictly contained in the ball there must be a bit of room for manoeuvre.

These are my thoughts on this problem. I'm not sure if they're correct or what the question is asking, and if they are I don't know how to write them more formally?

Thank you :)
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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Homework Statement



R>0, let K be a closed subset of C such that K [tex]\subset[/tex] BR(0) (so K is compact). Show that there exists 0 < r < R such that K[tex]\subset[/tex] Br(0).

Homework Equations





The Attempt at a Solution



Can I write BR(0) = {x[tex]\in[/tex]C : d(x,0) [tex]\leq[/tex]R}?
I know that a compact set is closed and bounded.
Is it something to do with us using [tex]\subset[/tex] and not [tex]\subseteq[/tex]?
As if it was [tex]\subseteq[/tex] then maybe K = BR(0) then there wouldn't be an r. But as K is strictly contained in the ball there must be a bit of room for manoeuvre.

These are my thoughts on this problem. I'm not sure if they're correct or what the question is asking, and if they are I don't know how to write them more formally?

Thank you :)
From the context of the problem, they must mean B_R(0) to be the open ball, d(x,0)<R. Not the closed ball. Otherwise, it wouldn't be true.
 
  • #3
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Oh that makes more sense.

I think I need to say something like you can always fit a smaller ball inside an open ball.

I remember in real analysis I showed for a non-empty bounded above set E and epsilon >0, there exists an x in E such that supE - epsilon < x [tex]\leq[/tex] sup E. Is it something like this?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Oh that makes more sense.

I think I need to say something like you can always fit a smaller ball inside an open ball.

I remember in real analysis I showed for a non-empty bounded above set E and epsilon >0, there exists an x in E such that supE - epsilon < x [tex]\leq[/tex] sup E. Is it something like this?
Something 'like' that could be made to work. But it's a lot easier if you use the definition of 'compact' and a covering of K by open balls.
 

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