Abstract
ABSTRACT:Bi‐neighborhood (bi‐nbhd) lattices are a generaliization of bitopological spaces which use a natural duality that associates nbhd filters with dual nbhd ideals to view a lattice as a “top down” structure, in which points are unnecessary and which uses ideals to determine dual nbhds. This leads to a definition of dual continuous functions between dual neighborhood lattices. The link between dual nbhd continuity and topological (top) continuity is established by proving that if f: X → Y is a one‐to‐one and onto function between top spaces X and Y, then f is top continuous if and only if the direct image function is a dual nbhd continuous function mapping P(X), the power set of X, onto P(Y). In addition, bi‐nbhd lattices are used to generate examples of Urysohn collections and quasiproximities.
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