A quasi-locally constant function with given cluster sets (Q1987687)

In the present paper, a further study of cluster sets is carried out for functions defined over open subsets of a topological space. The following is the main result of this exposition. Theorem. Let \(X\) be a metrizable topological space, \(Y\) be a dense subspace of a metrizable compact space \(\overline{Y}\), \(D\) be an open dense subset of \(X\), \(L=X\setminus D\), and \(\Phi:L\to 2^{\overline{Y}}\) be a multifunction. Then, the following are equivalent: (1) There exists a quasi-locally constant function \(f:D\to Y\) such that \(\overline{f}(x)=\Phi(x)\), for each \(x\in L\) (2) \(\Phi\) is an upper continuous compact-valued multifunction. Further aspects occasioned by these developments are also being discussed.