TITLE: Characterization of cocompact group action
QUESTION [15 upvotes]: Wikipedia claims the following:

In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space or, equivalently, if there is a compact subset K of X such that 
  the image of K under the action of G covers X.

My question is: Isn't this wrong? It is evident that the existence of such a subset K ensures cocompactness, but I am doubting the other direction. How could one possibly choose K? Taking an arbitrary transversal (or its closure) does not work, and I do not see what else could be a candidate.
By the way: Wikipedia points to a specific page in the Handbook of Geometric Topology. This page, however, contains only the definition of a cocompact space, not the claimed equivalence.

REPLY [9 votes]: Here is a counterexample in which $X/G$ is also Hausdorff.
We give $X:=(\mathbb{N} \times \mathbb{Z}) \cup \{\infty\}$ a topology similar to the Arens-Fort topology. That is, each point $(m,n) \in \mathbb{N} \times \mathbb{Z}$ is isolated and basic neighborhoods of $\infty$ are of the form $$B_{f,k}:=\{\infty\} \cup \{(m,n) : n \geq f(m) \land m \geq k\},$$ where $f:\mathbb{N} \to \mathbb{Z}$ and $k \in \mathbb{N}$. 
We let $G:=\mathbb{Z}$ act on $X$ in the natural way: $g(m,n)=(m,n+g)$ and $g(\infty)=\infty$.
Any compact subset of $X$ is finite, so its orbit cannot cover $X$. On the other hand $X/G$ is just a convergent sequence (which is compact and Hausdorff).