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?