Prove a set is closed and bounded but not compact in metric space

  • Thread starter hhqqvn89
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  • #1
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Homework Statement



Let X be the integers with metric p(m,n)=1, except that p(n,n)=0. Show X is closed and bounded but not compact.

Homework Equations



I already check the metric requirement.

The Attempt at a Solution



I still haven't got any clue yet. Can anyone help me out?
 

Answers and Replies

  • #2
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What is your definition of compact? The typical definition is every open cover has a finite subcover. Thus one way you could prove this is to find just one open cover such that there is no finite subcover.
 
  • #3
The set you are describing is known as the discrete metric except with integers instead of arbitrary numbers.

To determine if it is closed, you need to know if it has all its limit points. So the question you need to ask is "does this set have limit points"? If it does, are they in the set, if it does not then it is vacuously closed.

To determine if it is bounded you just need to find a ball that encompasses all the numbers which in this case is easy.

Now for non-compactness I'll give you a hint. An open ball of radius greater than one with a center at any point contains the entire set. Find a ball that does not do this then create a ball around each point.
 

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