I've been studying the theorems that establishes that , in a $T2$ space, every locally compact space is locally closed and every locally closed subset of a locally compact space is locally compact. My definition of locally compact space is the third one stated by Wikipedia: a space is locally compact if every point has a local base of compact neighbourhoods.
Now it's time to find counterexamples for the non-$T2$ case. I've already found than in right-order topology over reals the set $\{0\}$ is locally compact but not locally closed. It also can be proved, without using $T2$, that every closed or open subspace of locally compact space is locally compact. What I'm missing is an example of locally closed subset of a locally compact space that is not locally compact. Do you know one ?
An open subset of a locally compact space is always locally compact. This is almost trivial from the definition: if $U\subseteq X$ is open and $x\in U$, then $U$ is a neighborhood of $x$ so any local base for $X$ at $x$ can be restricted to a local base for $U$ at $x$ by just taking the elements of the local base that are contained in $U$. In particular, if there is a local base of compact neighborhoods in $X$, there is also one in $U$.
It then follows immediately that a locally closed subset of a locally compact space is locally compact: a locally closed subset is just a relatively closed subset of an open subset.