A note on Levine's decomposition of continuity
type: Item
instance of: scholarly article
title: A note on Levine's decomposition of continuity
zbMATH Open document ID: 0726.54009
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Levine's decomposition of continuity for an arbitrary function between arbitrary topological spaces is strengthened. A function \(f:X\to Y\) is weakly continuous if \(f^{-1}(V)\subset Int f^{-1}(Cl V)\) for each open set \(V\subset Y\) and f is \(w^*.c\). if \(f^{-1}(Bd V)\) is closed for each open \(V\subset Y\) where Int, Cl and Bd are the interior, closure and boundary operators respectively. Levine proved that a function is continuous if and only if it is both weakly continuous and \(w^*.c.\) A function \(f: X\to Y\) is weakly \(\alpha\)-continuous if and only if \(f^{-1}(V)\subset Int Cl Int f^{-1}(Cl V)\) for each open set \(V\subset Y\) and f is locally \(w^*.c\). if there is an open basis \({\mathcal B}\) for the topology on Y such that \(f^{-1}(Bd V)\) is closed for each \(V\in {\mathcal B}\). In this paper the following theorem is proved: A function \(f:X\to Y\) is continuous if and only if it is both weakly \(\alpha\)-continuous and locally \(w^*.c.\) Also some results in the literature are slightly improved.
Mathematics Subject Classification ID: 54C08
zbMATH DE Number: 4199005
zbMATH Keywords: Levine's decomposition of continuity
author: David A. Rose
reviewed by: Ľubica Holá
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4199005