Proving every compact space is locally compact.

Question

A topological space is said to be locally compact if each point $x\in X$ has at least one neighbourhood which is compact. Prove every compact space is locally compact.

I thought this problem was trivial but I am not sure. If I consider $(X,\tau)$ to be compact topological space then every $x\in X$ has X as neighbourhood which implies $(X,\tau)$ is locally compact.

Question:

Is this reasoning right?

Thanks in advance!

Answer 1

Yes, this reasoning is correct. $X$ itself is always a neighbourhood of any of its points.

Note however that there are many, many different variants of "locally compact" definition. All of them are equivalent for Hausdorff spaces, but in non-Hausdorff case you need to be extra careful about the definition you are working with.

So your reasoning is correct for this particular definition. But it is not necessarily true for other definitions.