Date: Nov 17, 2012 5:10 AM
Author: Achimota
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,