Talk:Pretopological space

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[edit] Dubious

I stuck a dubious tag on one of the assertions. The book in front of me states that the collection of open sets generated by the praclosure operator is a full-fledged, real topology, and not something less than that. Unless I flubbed something, this seems easy enough to prove. So I don't understand why praclosure leads to something less than a full topology. Am I missing something? (And, in general, I find this article confusing, and that's no help). linas 02:26, 25 November 2006 (UTC)

See comment at Talk:Praclosure operator. Those axioms don't give you binary unions of open sets. So I think you must be misquoting: generated by must mean taking all possible unions, first. You seem to think 'generated by' is innocuous. So I'm taking your tag down. Charles Matthews 15:01, 26 November 2006 (UTC)

I replied at that talk page, which now contains a proof of the main lemma needed to show that binary unions of open sets are open. So I still find this article dubious. linas 04:06, 28 November 2006 (UTC)

I cut the following out of the article:

In order to define a topology on X, the closure operator must also be idempotent; that is, it must satisfy for all subsets A of X:

cl (cl (A)) = cl (A).

For explanation why this is a necessary and sufficient condition, see Kuratowski closure axioms.

Complaints: 1) article on Kuratowski closure axioms fails to explain nec & suff. (as do all three of my books on topology). 2) the talk page of praclosure now contains a proof that the praclosure can generate a topology. In that topology, praclosure is idempotent on closed sets (that is how a closed set is defined).

Proposed solution: the topology generated by the praclosure is a topology, which can be quite strange, but not the usual topology; and it seems to me that the "usual" topology does require the fourth, idempotency, axiom. Does this now all make sense? linas 14:18, 28 November 2006 (UTC)