I was looking at the wikipedia definition for local compactness, and it gives various definitions, I am interested in these two:
definition A:
We say a topological space $X$ is locally compact if every point $x$ has a compact neighbourhood.
definition B:
We say a topological space $X$ is locally compact if every point $x$ has a compact local base.
The definition of a compact neighborhood of $x$ is a neighborhood of $x$ that is contained inside a compact set.
The definition of a compact local base of $x$ is a local base consisting of compact neighborhoods.
I think definition $A$ implies definition $B$:
Suppose $x\subseteq U \subseteq K$ is a compact neighbourhood of $x$. Then we can find a compact basis of $x$ by considering the neighbourhoods $U_\alpha \cap U$ with $\alpha \in A$ where the $U_\alpha$ are any basis of $x$.
Is this correct? The wikipedia page says they aren't equivalent so I am confused.
Best Answer
Here is an example of a compact $T_1$-space that is locally compact A but not locally compact B, where you want the neighbourhoods themselves to be compact, rather than just contained in some compact set. Take $\mathbb{Q}\cup\{\pi\}$ topologized as follows: rational numbers get their normal neighbourhoods; the neighbourhoods of $\pi$ are the co-finite sets. This space is locally compact A because the whole space is compact. It is not locally compact B, because no rational point has a neighbourhood that is compact in its own right that is disjoint from $\{\pi\}$.