```Date: Nov 17, 2012 11:36 AM
Author: Frederick Williams
Subject: Re: definition of closure in topological space question

William Elliot wrote:> > 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.cl(A) 2. should read  cl(A) = {x : for each neighbourhood N of x,                N intersect A =/= emptyset}.A neighbourhood of x is an element of a complete system ofneighbourhoods of x, denoted N_x.  A complete system of neighbourhoodsof x in X satisfiesFor all x in X, N_x =/= emptyset;For all x in X and N in N_x, x in N;For all x in X and N in N_x, if M superset N then M in N_x;For all x in X and N, M in N_x, N intersect M in N_x;For all x in X and N in N_x, there is an M in N_x such that M subset N  and M in N_y for each y in M.A subset O of X is open if O is a neighbourhood of each x in O.  Thus"neighbourhood spaces" and topological spaces with the usual open setaxioms are equivalent.With those definitions, cl version 1 and cl version 2 are (as one wouldexpect) equivalent.  I know nothing about these things, but I just don'twant the OP to be confused.[neighbourhood = neighborhood]-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
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