In message <Pine.NEB.email@example.com>, William Elliot <firstname.lastname@example.org> 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.)