Vol. 76, No. 1, 1978

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Neighbornets

Heikki J. K. Junnila

Vol. 76 (1978), No. 1, 83–108
Abstract

By a “neighbornet” of a topological space X we mean a binary relation V on X such that for each x X, V {x} is a neighborhood of x; thus a neighbornet is, in effect, an assignment of neighborhoods to the points of X. Such neighborhood assignments and the corresponding relations have been in use since the beginning of the study of general topology, at first in the theory of metric spaces and later in the theory of uniform spaces; in the last twenty or thirty years they have been used in connection with spaces defined by covering axioms and with various generalizations of metric and uniform spaces. Even though the concept of a neighbornet is not new, neighbornets have mostly been considered as tools, not as objects of intrinsic interest. With this paper we hope to show that neighbornets deserve to be studied also on their own. We shall show, for example, that from simple properties of neighbornets of semi-stratifiable spaces the solution follows easily to J. Ceder’s problem, whether all M3-spaces are M2-spaces.

Mathematical Subject Classification 2000
Primary: 54E15
Milestones
Received: 16 May 1977
Revised: 5 August 1977
Published: 1 May 1978
Authors
Heikki J. K. Junnila