Locally compact

  • Thread starter winmath
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hi.. how can we say a compact space automatically a locally compact? how subspace Q of rational numbers is not locally compact? am not able to understand these.. can anyone help me?
 

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  • #2
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Locally compact is in a certain sense a weaker assertion than compactness. Here is a hint to show that Q is not locally compact. Take a point x and define some neighbourhood. Then show that this neighbourhood is not contained in a compact subspace. To do so consider an irrational point in the neighbourhood. Also as a further hint use limit point compactness. This is one way to prove it. Let me know if you are still stuck.
 
  • #3
Landau
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Locally compact is in a certain sense a weaker assertion than compactness.
Why do you say "in a certain sense"? It's just weaker; locally compact means: every point has a compact neighbourhood. If the whole space is compact, for any point you just take the whole space as compact neighborhood.
how subspace Q of rational numbers is not locally compact?
Just try to find a compact neighborhood of q\in Q.
 
  • #4
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Landau; Thanks for the correction. Locally compactness is just weaker. I don't know what I was thinking.
 

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