```Date: Nov 17, 2012 9:32 PM
Author: William Elliot
Subject: Re: definition of closure in topological space question

On Sat, 17 Nov 2012, Frederick Williams wrote:> 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}.> No, that's so much unlike 2, that it can't be a correction of 2.That's the basic equivalent (as you show below) definition of cl A.OP is neither confused nor incorrect about 2.As he indicated, it's a metric space definition and in fact, in any metric space, 1 and 2 are equivalent.Are they equivalent in normal T1 spaces?> A neighbourhood of x is an element of a complete system of> neighbourhoods of x, denoted N_x.  A complete system of neighbourhoods> of x in X satisfies> > For 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 set> axioms are equivalent.> > With those definitions, cl version 1 and cl version 2 are (as one would> expect) equivalent.  I know nothing about these things, but I just don't> want the OP to be confused.> > [neighbourhood = neighborhood]
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