```Date: Nov 17, 2012 11:05 PM
Author: David Hartley
Subject: Re: definition of closure in topological space question

In message <Pine.NEB.4.64.1211171822240.21847@panix1.panix.com>, William 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.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.)-- David Hartley
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