Basic properties of regular-closed functions

Abstract

A function f:X→Y is defined to be regular-closed if for each regular-closed A⊂X, f(A) is closed in Y. Numerous theorems are presented which give properties of such functions as well as sufficient conditions for a function to be regular-closed. Comparisons are also made between regular-closed functions and certain other types of non-continuous functions. A sample of the theorems proved in this paper would be as follows: Theorem. A function f:X→Y is regular closed if and only if for each open or regularclosed A⊂X, cl(f(A))⊂f(cl(A)). Theorem. Normality is preserved under continuous regular-closed surjections. Theorem. Let f:X→Y be a function such that the induced graph function is almostcontinuous and almost-open. Let X be H-closed and Y Hausdorff. Then f is regular-closed. © 1978 Springer.