Study of families of monotone continuous functions on Tychonoff spaces
The paper deals with continuous (real-valued) monotone functions on an (infinite) connected Tychonoff space \(X\), where monotone means that the preimage of any point is connected. The class of all such functions on \(X\) is denoted by \(CM(X)\). The paper is a selfcontained text reporting on the ``state of the art'' on the subject and presenting solutions to many interesting problems of which we only cite a few. The following two situations are investigated, the case where the class \(CM(X)\) separates points (\(X\) is \(m\)-functionally separated) and the case where \(CM(X)\) contains only constant functions (\(X\) is functionally \(m\) singular). Necessary and/or sufficient conditions on a space \(X\) are given such that the space is functionally \(m\) separated (functionally \(m\) singular). Special attention goes to connected linearly ordered topological spaces which are shown to be functionally \(m\) separated and to compact metric spaces for which easy characterizations for both functionally \(m\) separatedness and functionally \(m\) singularity are formulated. Another important issue in the paper is the study of \(CM(X)\) as a subspace of \(C(X)\). Here \(C(X)\) is endowed with uniform convergence and with pointwise convergence. Important results are the following: \(CM(X)\) is nowhere dense in \(C(X)\) with uniform convergence. If \(X\) is normal and countably compact, then \(CM(X)\) is also closed in it. If \(X\) is a countable union of spaces \(X_i\) that are either a path or a point then \(CM(X)\) is closed and nowhere dense in \(C(X)\) with pointwise convergence. For \(X\) a countable union of compact connected linearly ordered spaces, pointwise convergence on \(CM(X)\) is shown to coincide with uniform convergence. Finally topological properties for \(CM(X)\) with pointwise convergence are studied, such as metrizability, separability and sigma compactness.
scientific article; zbMATH DE number 5319783