Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: definition of closure in topological space question
Replies: 11   Last Post: Nov 20, 2012 3:46 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
William Elliot

Posts: 1,600
Registered: 1/8/12
Re: definition of closure in topological space question
Posted: Nov 17, 2012 5:51 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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No wonder it's wrong; it's way out of date.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




Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.