Abstract
ABSTRACT: 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). T1‐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 T1‐nbhd lattices is continuous iff it preserves the limit of convergent nets.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
