This thesis is centered on the following question: Given an abstract group action by an Abelian group G on a set X, when is there a compact Hausdorff topology on X such that the group action is continuous? If such a topology exists, we call the group action compact-realizable.

We show that if G is a locally-compact group, a necessary condition for a G-action to be compact-realizable, is that the image of X under the stabilizer ma...