TITLE: Action that is Bourbaki proper but not Palais proper
QUESTION [14 upvotes]: 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).

REPLY [2 votes]: Here is another example: Consider a compact finite dimensional smooth manifold $M$, and let $\text{Met}(M)$ be the space of all smooth Riemannian metrics on $M$,
an open subset in the Frechet space $\Gamma(S^2T^*M)$. Then the regular Frechet Lie group $\text{Diff}(M)$ of all diffeomorphisms of $M$ acts smoothly on $\text{Met}(M)$ by pullback. The quotient space $\text{Met}(M)/\text{Diff}(M)$ is Hausdorff in the quotient topology. The action is free on generic Riemannian metrics, and is the isometry group of a metric $g$ in general, which is a compact Lie group in general, since $M$ is compact. Thus this action is B-proper, but not P-proper, since the group $\text{Diff}(M)$ is not locally compact.