For Hausdorff spaces the following are equivalent:
- Every point admits a compact local base.
- Every point admits a compact neighborhood.
- Every point admits a precompact neighborhood.
- Every point admits a precompact open neighborhood.
(Among these the local compactness is the usually applied one.)
For non-Hausdorff spaces:
What are examples where neither of them are equivalent?
What situations make the latter two become important?
(I'm just wondering as these are almost always taken as definition.)
The one-point compactification $\Bbb Q^*$ of $\Bbb Q$ is compact, so every point has a compact nbhd and a precompact open nbhd, but no point of $\Bbb Q$ has a local base of compact sets in $\Bbb Q^*$: $\Bbb Q$ is open in $\Bbb Q^*$, but every compact nbhd of a point of $\Bbb Q$ contains the point at infinity. Thus, $\Bbb Q^*$ satisfies your second and third conditions but not your first.
Let $\tau=\{\varnothing,\Bbb R\}\cup\{(x,\to):x\in\Bbb R\}$; $\tau$ is a topology on $\Bbb R$, and the compact subsets of $\Bbb R$ in this topology are precisely the sets that are bounded below. Every point actually has a base of compact open nbhds, but no point has a compact closed nbhd: the only closed set with non-empty interior is $\Bbb R$ itself, and it’s not compact. Thus, $\langle\Bbb R,\tau\rangle$ satisfies your first and second conditions but not your third.
These examples show that your first and third conditions are independent of each other. Each of them implies your second condition, so it only remains to find a space that satisfies your second condition but not your first or third. Let $\tau'=\{\varnothing\}\cup\{U\subseteq\Bbb N:0\in U\}$; $\tau'$ is a topology on $\Bbb N$. For each $n\in\Bbb N$ the set $\{n,0\}$ is a compact nbhd of $n$. However, $0$ has no precompact open nbhd: $\operatorname{cl}_{\tau'}\{0\}=\Bbb N$, which is not compact.
I can’t give you a specific answer to your last question: I’ve never had much occasion to work with local compactness in non-Hausdorff spaces. In general, though, it’s nice to know exactly what property you’re using when you prove something — for instance, whether you’re using a local base of compact sets, or just a single compact nbhd.