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]