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Re: definition of closure in topological space question
Posted:
Nov 17, 2012 11:05 PM
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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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