Dan
Posts:
3
Registered:
6/28/07
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Re: definition of closure in topological space question
Posted:
Nov 20, 2012 3:43 PM
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Thank you everyone for your replies to this post :)
-- OP
On Saturday, November 17, 2012 6:10:00 PM UTC+8, Daniel J. Greenhoe wrote:
> Closure in topological space is defined using at least two different ways in the literature:
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> 1. cl(A) is the intersection of all closed sets containing A.
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> 2. cl(A) is the intersection of all neighborhoods containing A, where a neighborhood is any set containing an open set (an element of the topology).
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> Examples of authors who use 1 include Kelley, Munkres, Thron, and McCarty.
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> Examples of authors who use 2 include Mendelson and Aliprantis & Burkinshaw.
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> My question is, one definition considered to be more "standard" than the other (from my very limited survey, 1 might seem more standard).
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> Aliprantis/Burkinshaw hints that 2 is influenced by metric space theory.
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> I might guess that there are other definitions possible (hence the "Kuratowski closure axioms"?)
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> Pointers to good references are especially appreciated.
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> Many thanks in advance,
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> Dan
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