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.
