Below *locally compact* spaces are assumed to be *Hausdorff*. The following is essentially a distillate of results from Bourbaki's *Topologie Générale*, Chapitres II and III.

**Definition.** A continuous function $f: X \to Y$ is called *proper* if $f$ maps closed sets to closed sets and $f^{-1}(K)$ is compact for all compact $K \subset Y$.

**Remark.** If $X$ is Hausdorff and $Y$ is locally compact then a continuous function $f: X \to Y$ is proper if and only if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Moreover, $X$ must be locally compact.

To see this, cover $Y$ with open and relatively compact sets $U_{\alpha}$. Then $f^{-1}(U_{\alpha})$ is an open covering of $X$ by relatively compact sets, hence $X$ is locally compact. If $F \subset X$ is closed then $f(F)$ is closed. Indeed, if $(y_{n}) \subset f(F)$ is a net converging to $y$, then we may assume that all $y_{n}$ are in a compact neighborhood $K$ of $y$. Pick a pre-image $x_{n}$ of each $y_{n} \in f^{-1}(K)$, which is compact by assumption. If $x_{i} \to x \in f^{-1}(K)$ is a convergent subnet of $(x_{n})$ then $(f(x_{i}))$ is a subnet of $(y_{n})$, hence $f(x) = y$ by continuity and thus $y \in f(F)$.

**Remark.** In the definition of properness it would suffice to require that $f$ is closed and $f^{-1}(y)$ is compact for all $y \in Y$, but the definition above is good enough for the present purposes.

**Definition.** Let $G$ be a topological group acting continuously on a topological space $X$. The action is called *proper* if the map $\rho: G \times X \to X \times X$ given by $(g,x) \mapsto (x,gx)$ is proper.

**Proposition.** If $G$ acts properly on $X$ then $X/G$ is Hausdorff. In particular, each orbit $Gx$ is closed. The stabilizer $G_{x}$ of each point is compact and the map $G/G_{x} \to Gx$ is a homeomorphism. Moreover, if $G$ is Hausdorff then so is $X$.

*Proof.* Indeed, the orbit equivalence relation is the image of $\rho$, hence it is closed. Since the projection $X \to X/G$ is open, this implies that $X/G$ is Hausdorff. Since the pre-image of the point $[x]$ in $X/G$ is its orbit $Gx$, we see that orbits are closed. The stabilizer $G_{x}$ of a point $x$ is the projection of $\rho^{-1}(x,x)$ to $G$, hence it is compact. The map $G/G_{x} \to Gx$ is proper and $1$-to-$1$, hence a homeomorphism. Finally, if $G$ is Hausdorff, then $\{e\} \times X \subset G \times X$ is closed and therefore the diagonal $\Delta_{X} = \rho(\{e\} \times X)$ of $X \times X$ is closed, hence $X$ is Hausdorff.

**Exercise.** Let $G$ be a Hausdorff topological group acting properly on a locally compact space $X$. Then $G$ and $X/G$ are both locally compact. If $X$ is compact Hausdorff then so are $G$ and $X/G$.

Replace *finite* by *compact* in Type A and Type B. Then we have the following implications for a continuous action:

Proper $\Longrightarrow$ Type A, the converse holds if both $G$ and $X$ are locally compact.

Type A $\Longrightarrow$ Type B.

Let $K \subset X$ be compact. Then $K \times K \subset X \times X$ is compact. Thus, if the action is of type A, then $\rho^{-1}(K \times K) = \{(g,x) \in G \times X\,:\,(x,gx) \in K \times K\} \subset G \times X$ is compact. The projection of this set to $G$ is compact and consists precisely of the $g \in G$ for which $K \cap gK \neq \emptyset$.

Type B $\Longrightarrow$ Type A if $X$ is Hausdorff.

We have to show that $\rho^{-1}(L)$ is compact for every compact $L \subset X \times X$. Let $K$ be the union of the two projections of $L$. Then $(g,x) \in \rho^{-1}(K \times K)$ is equivalent to $x \in K \cap gK$. Since $\rho^{-1}(K \times K)$ is compact and $\rho^{-1}(L)$ is a closed subset of $\rho^{-1}(K \times K)$, we have that $\rho^{-1}(L)$ is compact.

**Corollary.** If $G$ and $X$ are locally compact, properness, Type A and Type B are all equivalent.

Let me now show that in the locally compact setting properness is equivalent to a refinement of Type C:

**Proposition.** Let $G$ and $X$ be locally compact and assume that $G$ acts continuously on $X$. The following are equivalent:

- The action is proper.
- For all $x,y \in X$ there are open neighborhoods $U_{x}, U_{y} \subset X$ of $x$ and $y$ such that $C = \{g \in G\,:\,gU_x \cap U_{y} \neq \emptyset \}$ is relatively compact.

*Proof.* $1.$ implies $2.$ Let
$K_{x}$ and $K_{y}$
be compact neighborhoods of $x$ and $y$. Then the set $\rho^{-1}(K_{x} \times K_{y})$ is compact and its projection to $G$ contains $C$ and is compact. Now let $U_{x}$ and $U_{y}$ be the interiors of $K_{x}$ and $K_{y}$.

$2.$ implies $1$. Let $K \subset X \times X$ be compact. We want to show that $\rho^{-1}(K)$ is compact as well. Let $(g_{n},x_{n})$ be a universal net in $\rho^{-1}(K)$. Then $(x_{n},g_{n}x_{n})$ is a universal net in $K$ and hence converges to some $(x,y) \in K$. Let $U_{x}, U_{y}$ and $C$ be as in $2.$. Then $(x_{n},g_{n}x_{n}) \in U_{x} \times U_{y}$ eventually and thus also $(g_{n}) \subset C$ eventually. Since $(g_{n})$ is universal and $C$ is relatively compact, $(g_{n})$ converges to some $g \in G$. Hence $(g_{n},x_{n})$ converges to $(g,x) \in \rho^{-1}(K)$.

**Example.**
To see that Type C is weaker than properness, consider $A = \begin{pmatrix} 2 & 0 \\ 0 & 2^{-1} \end{pmatrix}$ and the action of $\mathbb{Z}$ on $\mathbb{R}^{2} \smallsetminus \{0\}$ given by $n \cdot x = A^{n} x$. For instance for $x = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $y = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ and all neighborhoods $U_{x} \ni x$ and $U_{y} \ni y$ the set $\{n \in \mathbb{Z}\,:\, U_{x} \cap n \cdot U_{y} \neq \emptyset \}$ is infinite. Thus this action isn't proper. On the other hand, it is easy to see that it is of Type C.

**Remark.**
The previous example shows that properness of an action is *not* a local property.

**Exercise.**
If the action of a locally compact group $G$ on a locally compact space $X$ is of type C and $X/G$ is Hausdorff then it is proper.

To finish this discussion, it is evident that an action of type C is also of type E, hence type E is also weaker than properness. Finally, a trivial action is of type D, hence this property has nothing to do with properness.

Here are some references:

I've followed Bourbaki, Topologie Générale, Ch. III, in terminology, and the proofs I've given are variants of Bourbaki's. I happen to like Koszul's *Lectures on groups of transformations*. If you're looking for a more pedestrian approach, you can find the most important facts in Lee's *Introduction to topological manifolds*.