Closed relations and equivalence classes of quasicontinuous functions
Let \((X,\tau)\) be a topological space. A set \(A\) of \(X\) is said to be quasi-open (or semi-open) if \(A\subset\text{Cl}(\text{Int}(A))\). A function \(f: (X,\tau)\to (Y,\sigma)\) is said to be quasicontinuous (or semicontinuous) if \(f^{-1}(V)\) is quasi-open (semi-open) for every open set \(V\) of \(Y\). If \(f\) is continuous then \(C_f= \{x\in X\mid f\) is continuous at \(x\}\). Let \(X\) and \(Y\) be compact metric spaces. A relation \(F: X\to Y\) is a subset of \(X\times Y\) such that \(F(x)= \{y\in Y\mid (x, y)\in F\}\) is nonempty. \(F\) is closed if it is closed as a subset of \(X\times Y\) in the product topology. The graph of any function is an example of a relation. We denote by \(\overline f= \text{Cl}(f)\) the closure of the graph of \(f\) in the product topology of \(X\times Y\). In this paper the authors introduces a notion of equivalence of two functions in this way: Definition. We say that the functions \(f,g: X\to Y\) are equivalent \((f\sim g)\) if \(\overline f=\overline g\) in \(X\times Y\). For quasicontinuous functions the following is proved: Theorem. Let \(f,g: X\to Y\) be quasicontinuous functions. Then we have: (i) \(f\sim g\) if and only if \(C_f= C_g\) and \(f\,|\,C_f= g\,|\,C_g\). (ii) Let \(h: X\to Y\). If \(h\subset\overline f\), then \(h\) is quasicontinuous and \(h\sim f\). (iii) \(\overline f=\bigcup_{h\sim f} h\). If \(h: X\to Y\) is a function (not necessarily a quasicontinuous function) it is proved in Theorem 4 that there exists a quasicontinuous function \(f: X\to Y\) such that \(f\subset\overline h\). Corollary 7 of Theorem 4 is a new equivalent definition for quasicontinuous function: A function \(f: X\to Y\) is quasicontinuous iff \(f\) has the property that \(g\subset\overline f\Rightarrow f\subset\overline g\). The authors show that this notion of equivalence between functions is of use in dynamical systems. As a consequence of Theorem 2 it is proved in Proposition 11 that if \(X\) and \(Y\) are compact spaces and \(f: X\to Y\) is quasicontinuous, then there exists a Borel measurable \(g: X\to Y\) with \(g\sim f\). Also, the authors show that not every quasicontinuous function has an invariant measure, but under certain circumstances it is equivalent to a quasicontinuous function that does have an invariant measure (Proposition 14).
scientific article; zbMATH DE number 5071501