Let $(X,\tau)$ be a topological space. A subspace is called precompact if its closure is compact. Now, local compactness
Definition 1 (Munkres): Every point $x$ has an open neighbourhood $U_x$ contained in a compact set $C_x$.
Definition 2 (given in my class): Every point $x$ has a precompact open neighbourhood.
Question: If $f:X\rightarrow Y$ is continuous and open. Suppose $X$ is locally compact, is $f(X)$ locally compact?
It is easy to show that the answer is yes, under Definition 1. Could someone provide a counterexample under Definition 2?
(Note: The two definitions are equivalent if we assume that the spaces are Hausdorff, so any counterexample must come from a non Hausdorff setting. That's my only lead.)
