When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex Functions-Jones, Three Dimensional Geometry and Topology- Thurston). Therefore, it will be clarified, if we write these definitions separately, and see which is stronger or which are equivalent? I will name them as "Type A", "Type B"..)

Let $X$ be a topological space and $G$ be a group acting on $X$.

Definition 1: The action is of "Type A" if the map $G \times X \rightarrow X \times X$, given by $(g,x)\mapsto (x,g.x)$ is proper, i.e. inverse image of any compact set under this map is compact.

Definition 2: The action is of "Type B" if for any compact set $K\subseteq X$, $K\cap g.K=\phi$ for all but finitely many $g\in G$.

Definition 3: The action is of "Type C" if for each $x\in X$ has an open neighbourhood $U$ such that $g.U\cap U=\phi$ for all but finitely many $g\in G$.

Definition 4: The action is of "Type D" if for each $x\in X$, there exist an open neighbourhood $U$ of $x$, such that $g.U\cap U\neq \phi$ for $g\in G$ implies $g.y=y$. g.x=x$.

Definition 5: The action is of "Type E" if each $x\in X$ has a neighbourhood $U$ such that the set {$ g\in G \colon g.x\in U $} is finite.

Q.1 Which type of actions imply which other type of action?

Q.2 If $X$ is Hausdorff, then under which type action, the quotient $X/G$ is Hausdorff?

(These are required, when studying action of a group on a compact Riemann surface, its quotient, whether quotient map is branched or unbranched, etc,)

(This question may be not applicable to post for MO; but when reading a paper related to enumeration of equivalent coverings of a space, with given (finite) transformation group, I came across this notion, and when looked into details, the different definitions puzzled. PARDON !!) puzzled.)

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex Functions-Jones, Three Dimensional Geometry and Topology- Thurston). Therefore, it will be clarified, if we write these definitions separately, and see which is stronger or which are equivalent? I will name them as "Type A", "Type B"..)

Let $X$ be a topological space and $G$ be a group acting on $X$.

Definition 1: The action is of "Type A" if the map $G \times X \rightarrow X \times X$, given by $(g,x)\mapsto (x,g.x)$ is proper, i.e. inverse image of any compact set under this map is compact.

Definition 2: The action is of "Type B" if for any compact set $K\subseteq X$, $K\cap g.K=\phi$ for all but finitely many $g\in G$.

Definition 3: The action is of "Type C" if for each $x\in X$ has an open neighbourhood $U$ such that $g.U\cap U=\phi$ for all but finitely many $g\in G$.

Definition 4: The action is of "Type D" if for each $x\in X$, there exist an open neighbourhood $U$ of $x$, such that $g.U\cap U\neq \phi$ for $g\in G$ implies $g.y=y$.

Definition 5: The action is of "Type E" if each $x\in X$ has a neighbourhood $U$ such that the set {$ g\in G \colon g.x\in U $} is finite.

Q.1 Which type of actions imply which other type of action?

Q.2 If $X$ is Hausdorff, then under which type action, the quotient $X/G$ is Hausdorff?

(These are required, when studying action of a group on a compact Riemann surface, its quotient, whether quotient map is branched or unbranched, etc,)

(This question may be not applicable to post for MO; but when reading a paper related to enumeration of equivalent coverings of a space, with given (finite) transformation group, I came across this notion, and when looked into details, the different definitions puzzled. PARDON !!)