Help with Understanding Locally Compact Spaces & Subspaces

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Discussion Overview

The discussion revolves around the concepts of locally compact spaces and compact spaces, specifically addressing why a compact space is considered locally compact and exploring the local compactness of the subspace of rational numbers, Q. The scope includes theoretical understanding and clarification of definitions in topology.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how a compact space can be automatically considered locally compact and seeks clarification on the local compactness of the rational numbers.
  • Another participant suggests a method to demonstrate that Q is not locally compact by examining neighborhoods and their relation to compact subspaces, mentioning the role of irrational points.
  • A participant asserts that locally compact is a weaker assertion than compactness, emphasizing that locally compact means every point has a compact neighborhood.
  • A later reply acknowledges a correction regarding the nature of local compactness, affirming that it is indeed weaker than compactness.

Areas of Agreement / Disagreement

Participants express differing views on the nature of local compactness and its relationship to compactness. There is no consensus on the explanation of local compactness in relation to the rational numbers.

Contextual Notes

Some assumptions about the definitions of compactness and local compactness may not be fully articulated, and the discussion does not resolve the nuances of these concepts.

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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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.
 
abiyo said:
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.
winmath said:
how subspace Q of rational numbers is not locally compact?
Just try to find a compact neighborhood of q\in Q.
 
Landau; Thanks for the correction. Locally compactness is just weaker. I don't know what I was thinking.
 

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