This paper is a sequel to one entitled “Varieties of topological groups”. The variety V of topological groups is said to be full if it contains every group which is algebraically isomorphic to a group in V For any Tychonoff space
X, the free group Fof V on Xexists, is Hausdorff and disconnected, and has Xas a closed subset. Any subgroup of Fwhich is algebraically fully invariant is a closed subset of F. If Xis a compact Hausdorff space, then Fis normal. Let V be a full Schreier variety and Xa Tychonoff space, then all finitely generated subgroups of Fare free in V.
A β-variety V is one for which the free group of V on each compact Hausdorff space exists and is Hausdorff. For any β-variety V and Tychonoff space
X, the free group of V exists, is Hausdorff and has Xas a closed subset. A necessary and sufficient condition for V to be a β-variety is given.
The concept of a projective (topological) group of a variety V is introduced. The projective groups of V are shown to be precisely the summands of the free groups of V. A finitely generated Hausdorff projective group of a Schreier variety V is free in V.