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

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William Elliot

Posts: 1,233
Registered: 1/8/12
Re: definition of closure in topological space question
Posted: Nov 18, 2012 2:44 AM
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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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