I'm working with different definitions of proper action (Cartan, Bourbaki and Palais) and the relation between them. All the spaces I'm working with are $T_{3.5}$, the definitions are:

If $U$ and $V$ are subsets of a $G$-space $X$ then we say that $U$ is **thin relative** to $V$ if $\{g \in G \; : \; gU \cap V \neq \emptyset \}$ has compact closure in $G$. If $U$ is thin relative to itself then we say that $U$ is **thin**. A subset $S$ of a $G$-space $X$ is **small** if every point of $X$ has a neighborhood wich is thin relative to $S$.

Bourbaki proper. Let $X$ be a $G$-space. We say that the action is Bourbaki proper (B-proper) if the function $\delta: G \times X \to X \times X$, $\delta(g,x) = (x,gx)$, is perfect (i. e. closed and the fibers of points $\delta^{-1}((x,x))$ are compact). If $G$ is locally compact, the action on $X$ is Bourbaki proper if and only if for every $x, y \in X$ there are neighborhoods $V_x$ and $V_y$ in $X$ of $x, y$, that are thin relative.

Palais proper. Let $X$ be a $G$-space with $G$ locally compact. We say that the action is Palais proper (P-proper) if each point of $X$ has a small neighborhood.

I need and example of an action that is B-proper but not P-proper . Some facts that could help are that P-proper implies B-proper and if the action is P-proper then the orbit space $X/G$ is $T_{3.5}$, like $X$, but if the action is B-proper then $X/G$ is only $T_2$.

Also if $X$ is locally compact then the two definitions are equivalent. Then, to build the example we need a $G$-space with $X$ not locally compact but that satisfy the axiom $T_{3.5}$ (or $T_{3}$) whose orbit space is $T_2$ but not $T_{3.5}$ $(T_{3})$. Some examples of space that are $T_{3.5}$ but not locally compact (where I have been looking for the example) are the Moore plane (Niemytski plane), Sorgenfrey line and infinite dimensional vector spaces (I have been trying with Hilbert cube).