Topology and its Applications 156 (2008) 28–45 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Statistical convergence in topology Giuseppe Di Maio a,1 , Ljubiša D.R. Koˇ cinac b,∗,2 a Dipartimento di Matematica, Seconda Università di Napoli, Via Vivaldi 43, 81100 Caserta, Italy b Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia article info abstract Article history: Received 16 July 2007 Received in revised form 20 January 2008 Accepted 31 January 2008 MSC: primary: 54A20 secondary: 54B20, 54C35, 54D20, 40A05, 40A30, 40A99, 26A03, 11B05 Keywords: Asymptotic density Statistical convergence Statistical limit point Statistical cluster point Selection principles Function spaces Hyperspaces We introduce and investigate statistical convergence in topological and uniform spaces and show how this convergence can be applied to selection principles theory, function spaces and hyperspaces.  2008 Elsevier B.V. All rights reserved. 1. Introduction The notion of statistical convergence of sequences of real numbers was introduced by H. Fast in [13] and H. Steinhaus in [46] (see also [45]) and is based on the notion of asymptotic density of a set A ⊂ N [21,37,38]. However, the first idea of statistical convergence appeared (under the name almost convergence) in the first edition (Warsaw, 1935) of the celebrated monograph [49] of Zygmund. It should be also mentioned that the notion of statistical convergence has been considered, in other contexts, by several people like R.A. Bernstein, Z. Frolik, etc. Statistical convergence has several applications in different fields of mathematics: summability theory [5,14,15], number theory [12], trigonometric series [49], probability theory [17], measure theory [36], optimization [40] and approximation theory [18]. The statistical convergence was generalized to sequences in metric spaces (see, for instance, [34]). We introduce and study statistical convergence in topological and uniform spaces and offer some applications to selection principles theory, function spaces and hyperspaces. All spaces are assumed to be Hausdorff. Our topological terminology and notation is as * Corresponding author. E-mail addresses: giuseppe.dimaio@unina2.it (G. Di Maio), lkocinac@ptt.yu (L.D.R. Koˇ cinac). 1 Supported by MURST – PRA 2000. 2 Supported by MN RS. 0166-8641/$ – see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2008.01.015