Generalizations of boundedness, compactness and the tychonoff theorem

AbstractA restrictedness on a topological space X is a collection K of subsets of X, the elements of which are said to be restricted. X is locally restricted if every point has a restricted neighborhood.A type of restrictedness is a collection T of restrictednesses (not necessarily all on the same space); T(X) will denote the collection of members of T which are restrictednesses on the space X. X is called T-compact if for every R ϵ T(X), X is restricted if it is locally restricted.The Tychonoff Theorem is generalized in a form which gives a necessary and sufficient condition for a product of spaces to be T-compact for any given type T. Some applications are given, and several noteworthy types of restrictedness are introduced, including ones which display compactness, connectedness and H-closedness as special cases of a more general concept