Cluster sets and approximation properties of quasi-continuous and cliquish functions
A function \(f:X\rightarrow \mathbb R\) defined on a topological space \(X\) is said to be quasi-continuous (cliquish) if for each \(\epsilon > 0\), each point \(x\in X\) and each neighborhood \(U\) of \(x\) there is an open set \(V \subset U\) such that \(| f(x)-f(y)| < \epsilon\) for all \(y\in V\) (\(| f(y)-f(z)| < \epsilon\) for all \(y,z\in V\)). It is shown that any quasi-continuous (cliquish) real-valued function \(f\) on a topological space \(X\) can be uniformly approximated by a sequence \((\varphi_n)_{n\in\mathbb N}\) of semi-open (almost semi-open) step functions which are defined on a chain \((\mathcal P_n)_{n\in\mathbb N}\) of semi-open (almost semi-open) partitions \(\mathcal P_n =\{P_i^n:i\in I_n\}\). Here, for each \(n\), \(\mathcal P_{n+1}\) refines \(\mathcal P_n\), all elements of each \(\mathcal P_n\) are semi-open sets, and for each \(n\), \(\varphi_n\) is constant on the sets \(P_i^n\); an almost semi-open partition \(\mathcal P_n\) satisfies \(\cup\{intP: P \in \mathcal P_n\}\) is dense in \(X\).
scientific article; zbMATH DE number 2141351