On a Family of Functions Defined by the Boundary Operator

Abstract

For a topological space $X$, let $M(X,R)$ denote the family of all functions $f\in R^{X}$ such that $f(Fr(A))\subseteq Fr(f(A)).$ Let $N(X,R)$ denote the family of all continuous functions $f\in R^{X}$ such that $card(f^{-1}(c))=1$ for each $c\in \Biggl( \inf\limits_{x\in X}f(x),\sup\limits_{x\in X}f(x)\Biggr) .$ We show that $M(X,R)=N(X,R)$ if $X$ is a connected and locally connected Hausdorff space.