Date: Nov 17, 2012 5:10 AM
Subject: definition of closure in topological space question
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,