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.