```Date: Nov 18, 2012 2:44 AM
Author: William Elliot
Subject: Re: definition of closure in topological space question

On Sun, 18 Nov 2012, David Hartley wrote:> Elliot <marsh@panix.com> writes> > > > >   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).> ...> > 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.> > If A is open then it is a neighbourhood containing A, and so under 2, > cl(A) = A. How simple and direct.> That is not equivalent to the usual definition in any space which has an > open set which is not closed. In particular, it is only equivalent in a > T1 space if it is discrete.> > 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?
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