On the set of points of discontinuity for functions with closed graphs. (English) Zbl 0581.54008

Doboš, Jozef

Čas. Pěst. Mat. 110, 60-68 (1985).

For two topological spaces X and Y and any function \(f: X\to Y\) let \(D_ f\) denote the set of all points of discontinuity of f. This paper characterizes the set of all points of discontinuity of a real-valued function with closed graph defined on a perfectly normal space X, extending results of I. Baggs [Proc. Am. Math. Soc. 43, 439-442 (1974; Zbl 0281.54005)]. The main results are the following: Theorem 1. Let X be a perfectly normal topological space. Then \(A\subset X\) is closed and of the first category in X if and only if there exists a real- valued function f with closed graph defined on X such that \(D_ f=A\). Theorem 2. Let X be a Baire metric space. Then \(F\subset X\) is closed and nowhere dense in X if and only if there exists a real-valued function f with closed graph defined on X such that \(D_ f=F\). An example is given showing that the assumption ”X is perfectly normal” in Theorem 1 cannot be replaced by the assumption ”X is normal”.

Reviewer: Y.S.Yang

MSC:

54C10

Special maps on topological spaces (open, closed, perfect, etc.)

Keywords:

points of discontinuity of a real-valued function with closed graph; perfectly normal topological space; Baire metric space

Citations:

Zbl 0281.54005