Properties of generic and almost every mappings in

In a Polish group G, a property is said to hold for a generic g ∈ G if the set on which it does not hold is meager in G; a property holds for almost every (ae) g ∈ G if the set on which it does not hold is Haar null. In this paper we study the properties of generic and ae elements of the Baer–Specker Group Z Z , the space of all self-maps of Z . We find that with respect to most of the properties under consideration in this paper, the properties of a generic element and ae element are complementary. Our results are similar in flavor to results by Dougherty and Mycielski for the group S ∞ of all permutations of N .