Bi-Neighborhood Lattices
by
Frank Prokop
University of Wollongong

Bi-topological spaces examine the interaction between two topologies on the same set and allow one to view the set from two viewpoints and to examine the interrelationship between the imposed topological structures and the corresponding continuous functions. Bi-neighborhood (bi-nbhd) lattices give differing viewpoints of the same lattice structure but use a natural duality that associates nbhd filters with dual nbhd ideals to give two differing structural viewpoints of thesame `mathematical reality'; one as a `bottom up' structure, which may have points as a topological space and which uses filters to determine nbhds, and the other as a `top down' structure, in which points are unnecessary and which uses ideals to determine dual nbhds. Not only does each of the nbhd and dual nbhd structures have associated definitions of continuous functions, but also there is a kind of continuity in each direction between nbhd and dual nbhd 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).

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-65.