J. Math. Anal. Appl. 327 (2007) 142–154 www.elsevier.com/locate/jmaa A local version of the closed graph lemma of Kelley Corneliu Ursescu “Octav Mayer” Institute of Mathematics, Romanian Academy Ia¸ si Branch, 8 Copou blvd, Ia¸ si 700505, Romania Received 23 November 2005 Available online 11 May 2006 Submitted by M. Milman Abstract The paper concerns the local, metric version of a closed graph result of Kelley.  2006 Elsevier Inc. All rights reserved. Keywords: Multifunction openness; Multifunction near openness 1. Introduction In this paper, we derive a local, metric version of the metric-uniform result in [6, p. 202, Lemma 36]. To begin with, we rephrase that result by using the terminology of the consequent result in [14, p. 505, Theorem 2.1]. Let X and Y be topological spaces, let F : X → Y be a multifunction, and recall F is said to be open if for every open set U ⊆ X the set F(U) ⊆ Y is open. This global notion can be analyzed through a pointwise notion. The multifunction F is said to be open at a point (x,y) ∈ graph(F ) if for every neighborhood U of x the set F(U) is a neighborhood of y . Obviously, F is open if and only if it is open at every point of its graph. Now, the multifunction F is said to be nearly open at the point (x,y) if for every neighborhood U of x the set F(U) is a neighborhood of y . Here, S stands for the closure of the set S . This pointwise notion can be used to synthesize a global notion. The multifunction F is said to be nearly open if it is nearly open at every point of its graph. Obviously, F is nearly open if and only if for every open set U ⊆ X the set F(U) ⊆ Y is a neighborhood of F(U). Further openness and near openness notions can be devised in pairs, every openness implies the corresponding near openness, and moreover, the converse implication holds under appropri- E-mail address: corneliuursescu@yahoo.com. 0022-247X/$ – see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.04.005