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 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]

--

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