Help with Understanding Locally Compact Spaces & Subspaces

[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.

 

[blue_raver22]

Hello,

I can understand why you may be having trouble understanding locally compact spaces and subspaces. Let me try to explain it to you.

A compact space is a topological space in which every open cover has a finite subcover. This means that if we have a collection of open sets that covers the entire space, we can choose a finite number of them to still cover the space. In other words, compact spaces have a finite "size" in terms of open sets.

A locally compact space, on the other hand, is a topological space in which every point has a compact neighborhood. This means that for every point in the space, we can find a compact subset that contains that point and is also contained within an open set.

Now, to answer your first question, we can say that a compact space is automatically locally compact because every point in a compact space has a compact neighborhood. This is because, in a compact space, we can choose the entire space itself as a compact neighborhood for each point.

As for your second question, the subspace Q of rational numbers is not locally compact because not every point in Q has a compact neighborhood. For example, the point 0 does not have a compact neighborhood in Q because any open set containing 0 will also contain irrational numbers, which are not part of Q. Therefore, Q is not locally compact.

I hope this helps in understanding locally compact spaces and subspaces. If you have any further questions, please let me know.