
Re: definition of closure in topological space question
Posted:
Nov 17, 2012 5:51 AM


On Sat, 17 Nov 2012, Daniel J. Greenhoe wrote:
> Closure in topological space is defined using at least two different ways in the literature: > 1. cl(A) is the intersection of all closed sets containing A.
> 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). Those definitions aren't equivalent. Consider Sorgenfrey's two point space S = { 0,1 } with the topology { empty set, {0}, S }.
By 1, cl {0} = S while by 2, cl {0} = {0} isn't even a closed set.
> Examples of authors who use 1 include Kelley, Munkres, Thron, and McCarty. > Examples of authors who use 2 include Mendelson and Aliprantis & Burkinshaw. > My question is, one definition considered to be more "standard" than the > other (from my very limited survey, 1 might seem more standard).
Yes, 1 is the one to be used. 2 is bogus as I showed. > Aliprantis/Burkinshaw hints that 2 is influenced by metric space theory.
No wonder it's wrong; it's way out of date. > I might guess that there are other definitions possible (hence the > "Kuratowski closure axioms"?)
Yes and some theorems giving equivalent statements for closure.
> Pointers to good references are especially appreciated. >

