Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
David Hartley

Posts: 451
Registered: 12/13/04
Re: definition of closure in topological space question
Posted: Nov 18, 2012 10:44 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.