I was looking at the wikipedia definition for local compactness, and it gives various definitions, I am interested in these two:
We say a topological space $X$ is locally compact if every point $x$ has a compact neighbourhood.
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.