Closed graphs and closed projections

Abstract

If X and Y are topological spaces, we say that the pair (X, Y) has the closed graph property (C.G.P.) if every function on A CX into Y with a closed graph G(f) in X X Y is continuous on A. We say that the pair (X, Y) has the closed projection property (C.P.P.) if the projection ri of X X Y onto X is a closed function, i.e. if r1C is closed for all closed subsets C of X X Y. If 7rw maps the closures of open sets onto closed sets, then we say that the pair (X, Y) has the regular closed projection property (R.C.P.P.). A space X is said to be H(i) if every open filter base on X has nonvoid adherence. If Y is compact, then (X, Y) has C.G.P. and C.P.P. for all spaces X. Both of these results are well known, e.g. see [2, pp. 228-229]. Also, a discussion of C.G.P. can be found in [10]. If Y is H(i), then (X, Y) has R.C.P.P. by [8, p. 136]. Closed graphs, closed projections, closed relations and the relation of these properties to various compactness conditions have been studied in [3], [5], [6], [7] and [8]. It is the purpose of this paper to investigate C.G.P., C.P.P. and R.C.P.P., and to elaborate on and extend some of the results in the papers mentioned above. We also show that the properties feeble compactness or light compactness, and R.C.P.P. are closely related. The closure of a set A will be denoted by A', and ri will denote the first projection mapping. Inclusion will be denoted by C and proper inclusion by C. The positive integers will be denoted by I. The author wishes to thank the referee for bringing several references to his attention, particularly [3 ], which seems to overlap somewhat with this paper, and for improving Theorem 5 and Corollary 6.