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A Minimal Sober Topology Is Always Scott
Author(s) -
MCCLUSKEY A.E.,
McCARTAN S.D.
Publication year - 1996
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.1749-6632.1996.tb49176.x
Subject(s) - axiom , network topology , topology (electrical circuits) , comparison of topologies , set (abstract data type) , lattice (music) , general topology , complete lattice , mathematics , separation axiom , computer science , extension topology , discrete mathematics , combinatorics , topological space , physics , geometry , universality (dynamical systems) , quantum mechanics , acoustics , programming language , operating system
Given the lattice of all topologies definable for an infinite set X, we employ he well‐developed contraction technique (of intersecting a given topology with a suitably chosen principal ultratopology) for solving many minimality problems. We confirm its potential in characterising and, where possible, identifying those topologies which are minimal with respect to certain separation axioms, notably that of sobriety and its conjunction with various other axioms. Finally, we offer an alternative description of each topologically established minimal structure in terms of the behavior of the naturally occurring specialization order and the intrin sic topology on the resulting partially‐ordered set.