Date: Nov 17, 2012 5:51 AM
Author: William Elliot
Subject: Re: definition of closure in topological space question

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No wonder it's wrong;  it's way out of date.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.
>