General formulations of some theorems of cluster sets
type: Item
instance of: scholarly article
title: General formulations of some theorems of cluster sets
zbMATH Open document ID: 1015.30015
author: A. K. Layek
published in: Real Analysis Exchange
publication date: 2003-01-05T00:00:00Z
review text: 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.
reviewed by: Indrajit Lahiri
Mathematics Subject Classification ID: 30D40
zbMATH DE Number: 1848810
zbMATH Keywords: grill
zbMATH Keywords: porous
zbMATH Keywords: countable
zbMATH Keywords: category
zbMATH Keywords: measure
zbMATH Keywords: Baire property
zbMATH Keywords: cluster set
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1848810