Some new separation axioms are introduced and studied. We also deal with maps having an extension to a homeomorphism between the Wallman compactifications of their domains and ranges.2000 Mathematics Subject Classification: 54D10, 54D35, 54F65, 18A40.
Introduction.Among the oldest separation axioms in topology there are three famous ones, T 0 , T 1 , and T 2 .The T 0 -axiom is usually credited to Kolmogoroff and the T 1 -axiom to Fréchet or Riesz (and spaces satisfying the axioms are sometimes called Kolmogoroff spaces, Fréchet spaces, or Riesz spaces, accordingly). The T 2 -axiom is included in the original list of axioms for a topology given by Hausdorff [10].We denote by Top the category of topological spaces with continuous maps as morphisms, and by Top i the full subcategory of Top whose object is T i -spaces. It is a part of the folklore of topology that Top i+1 is a reflective subcategory of Top i , for i = −1, 0, 1, with Top −1 = Top. Thus Top i is reflective in Top, for each i = 0, 1, 2 (see MacLane [17]). In other words, there is a universal T i -space for every topological space X; we denote it by T i (X). The assignment X T i (X) defines a functor T i from Top onto Top i , which is a left adjoint functor of the inclusion functor Top i Top.The first section of this paper is devoted to the characterization of morphisms in Top rendered invertible by the functor T 0 .Let X be a topological space. Then T i (X) is a T i -space; moreover, T i (X) may be a T i+1 -space. The second section deals with space X such that T i (X) is a T i+1 -space.
The main goal of this work is to deal with the Khalimsky digital topology and its application in segmentation. First, we begin by giving some theoretical results on Khalimsky topology, the one point compactification and separation axioms. Then, we present and discuss digital applications in imaging. More precisely, numerical results on segmentation, contours detection and skeletonization are proposed. We end this paper by some concluding remarks.
This paper deals with lattice-equivalence of topological spaces. We are concerned with two questions: the first one is when two topological spaces are lattice equivalent; the second one is what additional conditions have to be imposed on lattice equivalent spaces in order that they be homeomorphic. We give a contribution to the study of these questions. Many results of Thron [Lattice-equivalence of topological spaces, Duke Math. J. 29 (1962), 671-679] are recovered, clarified and commented.2000 AMS Classification: 54B30, 54D10, 54F65.
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