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No wonder it's wrong; it's way out of date.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