Abstract. This paper considers the problem of defining Cauchy sequence and completeness in quasi-pseudo-metric spaces. The definitions proposed allow versions of such classical theorems as the Baire Category Theorem, the Contraction Principle and Cantor's characterization of completeness to be formulated in the quasi-pseudo-metric setting.
In this paper we define a new property of functions between topological spaces which is the dual of Blumberg's original notion, in the sense that together they are equivalent to continuity. Thus we provide a new decomposition of continuity in Theorem 4 (iv) which is of some historical interest.In a recent paper [10] , Tong introduced the notion of an A-set in a topological space and the concept of A-continuity of functions between topological spaces. This enabled him to produce a new decomposition of continuity. In this paper we improve Tong's decomposition result and provide a decomposition of A-continuity.Let S be a subset of a topological space (X, τ ) . We denote the closure of S and the interior of S with respect to τ by clS and intS respectively.
The aim of this paper is to give a systematic development of grill Ntopological spaces and discuss a few properties of local function. We build a topology for the corresponding grill by using the local function. Furthermore, we investigate the properties of weak forms of open sets in the grill N -topological spaces and discuss the relationships between them.2010 MSC: 54A05; 54A99; 54C10.
ABSTPCT. In this paper we introduce and study three different notions of generalized continuity, namely LC-irresoluteness, LC-continuity and sub-LC-continuity. All three notions are defined by using the concept of a locally closed set.A
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