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Topic: definition of closure in topological space question
Replies: 11   Last Post: Nov 20, 2012 3:46 PM

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: definition of closure in topological space question
Posted: Nov 17, 2012 11:36 AM
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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

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