Why is every compact space locally compact?

[winmath]

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?

 

[abiyo]

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.

 

[Landau]

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.

 

[abiyo]

Landau; Thanks for the correction. Locally compactness is just weaker. I don't know what I was thinking.