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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: definition of closure in topological space question
Posted: Nov 17, 2012 5:51 AM
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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No wonder it's wrong; it's way out of date.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.

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