A cyclic element characterization of monotone normality (Q5940643)

A topological Hausdorff space \(X\) is said to be monotonically normal provided that there exists a function \(G\) which assigns, to each point \(x\in X\) and each open set \(U\) of \(X\) containing \(x\), an open set \(G(x, U)\) such thatNEWLINENEWLINENEWLINE(1) \(x\in G(x,U)\subset U\),NEWLINENEWLINENEWLINE(2) if \(U'\) is open and \(x\in U\subset U'\), then \(G(x, U)\subset G(x,U')\),NEWLINENEWLINENEWLINE(3) if \(x\) and \(y\) are distinct points of \(X\), then \(G(x,X\smallsetminus\{y\})\cap G(y,X\smallsetminus\{x\})= \emptyset\).NEWLINENEWLINENEWLINEA cyclic element \(C\) of a locally connected continuum \(X\) means a subcontinuum of \(X\) that is maximal with respect to the property that no point separates \(C\). The authors prove that a locally connected continuum \(X\) is monotonically normal if and only if each cyclic element of \(X\) is monotonically normal. The result is an improvement or an analog of several earlier results in the area, due to various authors.