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