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Lower Density Topologies a
Author(s) -
ROSE D. A.,
JANKOVIĆ D.,
HAMLETT T. R.
Publication year - 1993
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.1749-6632.1993.tb52533.x
Subject(s) - mathematics , baire space , measure (data warehouse) , operator (biology) , space (punctuation) , second countable space , ideal (ethics) , countable set , field (mathematics) , topology (electrical circuits) , pure mathematics , discrete mathematics , combinatorics , computer science , data mining , biochemistry , chemistry , philosophy , epistemology , repressor , transcription factor , gene , operating system
A bstract By considering lower density operators and their induced topologies in a general setting, some results of S. Scheinberg and E. Ľazarow et al . are unified and generalized. It is also shown that every σ‐finite complete measure space ( X, M, m ) has a lower density operator and that every such operator induces a topology making X a category measure space in the sense of J. C. Oxtoby, except that the measure need not be finite. One consequence is that category σ‐finite measure spaces must have the countable chain condition. Also, for every topological space ( X , |Gt ), there is a lower density operator on the σ‐field of sets having the property of Baire (relative to the σ‐ideal of meager sets). Further, in both the “measure” and “category” contexts, all induced lower density topologies have simple form. Finally, it is shown that the deep. J ‐density operator on the σ‐field of subsets of the real line having the property of Baire is not a lower density operator.