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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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 David Hartley Posts: 463 Registered: 12/13/04
Re: definition of closure in topological space question
Posted: Nov 18, 2012 10:44 AM

In message <Pine.NEB.4.64.1211172248550.8793@panix1.panix.com>, William
Elliot <marsh@panix.com> writes
>> Make it *closed* neighbourhoods of A in 2 and then it's equivalent to
>> usual closure in T1 normal spaces, even regular spaces. (Probably it's
>> equivalent if and only if the space is regular.)

>
>More than T1 is needed for by 2, within the cofinite reals, cl {0} = R.
>
>Can you show the equivalence for normal T1 spaces?

Let Cn(A) be the intersection of all closed neighbourhoods of A, (where
a closed neighbourhood is a closed set C such that there is an open set
U with A c= U c= C).

Claim.
A space X is regular iff Cn(A) = Cl(A) for every subset A of X.

If X is not regular, then there exists x e X and a (closed) subset A of
X such that x is not in A but every nbhd. of x meets every nbhd. of A.
But then x is in every closed nbhd. of A and so is in Cn(A). Hence Cn(A)
=/= Cl(A).

Conversely, if Cn(A) =/= Cl(A) for some A, then regularity fails at any
x in Cn(A) \ Cl(A).
--
David Hartley

Date Subject Author
11/17/12 Achimota
11/17/12 William Elliot
11/17/12 Frederick Williams
11/17/12 William Elliot
11/17/12 David Hartley
11/18/12 William Elliot
11/18/12 David Hartley
11/18/12 William Elliot
11/19/12 David Hartley
11/19/12 William Elliot
11/20/12 Dan
11/20/12 Achimota