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.