General formulations of some theorems of cluster sets (Q1852342)

A collection \(P\) of subsets of the real line \(\mathbb{R}\) ( respectively complex plane \(\mathbb{E}_{2}\)) is called a grill [cf. \textit{W. J. Thron}, Math. Ann. 206, 35-62 (1973; Zbl 0256.54015)] in \(\mathbb{R}\) (respectively \(\mathbb{E}_{2}\)) if (i) \(\emptyset \not \in P\), (ii) \(A\in P\) and \(A\subset B\) implies \(B\in P\) and (iii) \(A\cup B \in P\) implies either \(A\in P\) or \(B\in P\). Let \(f: \mathbb{R} \longrightarrow W\), where \(W\) is a topological space. Let \(P\) be a grill in \(\mathbb{R}\). For \( U\subset W\), set \( f^{-1} (U)= \{x : x\in \mathbb{R}, f(x)\cap U \neq \emptyset\}.\) Let \(f\) be an one or multivalued function. Then the right hand \(P\)-cluster set \(C_{P}^{+}(f,x)\) of \(f\) at \(x\in \mathbb{R}\) is the set of all \(w\in W\) such that for every open set \(U\) of \(W\) containing \(w\), \(f^{-1}(U)\cap (x,x+r)\in P\) for all \(r >0\). The definition of \(C_{P}^{-}(f,x)\), the left hand cluster set of \(f\) at \(x\) is analogous. It is proved in the paper that if \(W\) is second countable then \(C_{P}^{+}(f,x)= C_{P}^{-}(f,x)\) except at most a countable set of points \(x\) of \(\mathbb{R}\). Several known results follow as consequences of this result. Also some results on \(P\)-cluster sets in the complex plane \(\mathbb{E}_{2}\) are proved.