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A Step Beyond Topology
Author(s) -
PROKOP FRANK P.
Publication year - 1994
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1994.tb44136.x
Subject(s) - mathematics , power set , dual polyhedron , closure (psychology) , complete lattice , topological conjugacy , closed set , pure mathematics , generalization , modulus of continuity , topology (electrical circuits) , direct limit , limit (mathematics) , lattice (music) , function (biology) , topological space , discrete mathematics , set (abstract data type) , combinatorics , computer science , mathematical analysis , type (biology) , physics , universality (dynamical systems) , ecology , quantum mechanics , evolutionary biology , economics , acoustics , market economy , biology , programming language
The questions which motivated the development of neighborhood (nbhd) lattices as a generalization of topological (top) spaces are discussed. Nbhd systems, which are shown to be appropriate for characterizing continuity, are defined on ^‐semilattices, and are used to define open elements. The duals of nbhd systems are used to define closed elements in a lattice, independently of closure operators or complementation. In addition, the top continuity of a function f : X → Y is characterized in terms of the nbhd continuity of the direct image function mapping P(X) , the power set of X , into P(Y). T 1 ‐nbhd lattices are defined, independently of points. Finally, the relationship between continuity and convergence is established by proving that a residuated function between conditionally complete T 1 ‐nbhd lattices is continuous iff it preserves the limit of convergent nets.