Dan
Posts:
3
Registered:
6/28/07


Re: definition of closure in topological space question
Posted:
Nov 20, 2012 3:43 PM


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: > > 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). > > > > 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). > > > > Aliprantis/Burkinshaw hints that 2 is influenced by metric space theory. > > > > I might guess that there are other definitions possible (hence the "Kuratowski closure axioms"?) > > > > Pointers to good references are especially appreciated. > > > > Many thanks in advance, > > Dan

