On insertion of a continuous function

Recently E. P. Lane proved that if space X has the weak C-insertion property and satisfies another condition, then X has the strong C-insertion property. This paper establishes the converse of this result. In a recent paper, E. P. Lane [1] established two main results about the insertion of continuous functions. The second of these is: THEOREM 3.1. Let P, and P2 be C-properties and consider the following condition: (a) If g and f are functions on X such that g O} = Un .I F,, and (ii) for each n the sets A(f g, 2-) and Fn are completely separated. If X satisfies the weak C-insertion property for (P1, P2) and if X satisfies (a), then X satisfies the strong C-insertion property for (P1, P2). Conversely, if X satisfies the strong C-insertion property for (P1, P2) and f g satisfies the property P1 (actually P1 should have read PF), then X satisfies (a). This paper shows that the last sentence of the above theorem can be strengthened to read: Conversely, if X satisfies the strong C-insertion property for (P1, ,,P2), then X satisfies (a). Thus (a) is a necessary and sufficient condition for a space with the weak C-insertion property to have the strong C-insertion property. A property P defined relative to a real-valued function on a topological space is a C-property provided any constant function has property P and provided the sum of a function with property P and any continuous function also has property P. Let P1 and P2 be C-properties. A space X is said to have the weak C-insertion property for (P1, ,,P2) iff for any functions g and f on X such that g < f, g has property P1 and f has property P2, then there exists a continuous function h on X such that g < h < f. A space X is said to have the strong C-insertion property for (P1, P2) iff for any functions g andf on X such that g < f, g satisfies P1 andf satisfies P2, then there exists a continuous function h on X such that g < h < f and such that if g(x) < f(x) for any x in X, then g(x) < h(x) < f(x). If f is a real-valued function defined on a space X and if {x/f(x) < r) C A(f, r) c {x/f(x) < r}, Received by the editors October 14, 1979 and, in revised form, January 31, 1980. AMS (MOS) subject classifications (1970). Primary 54C30; Secondary 54C05.