Talk:Preclosure operator

Topology or not

The collection of all open sets generated by the praclosure operator is a topology. Really? If you mean that this collection of 'open sets' is automatically closed under unions, I don't see why. It will be closed under binary intersection, from the axioms. Charles Matthews 14:49, 26 November 2006 (UTC)

That open sets are closed under arbitrary union is equivalent to showing that closed sets are closed under arbitrary intersection, and the latter is easier to prove. So let (C_i) be a collection of closed sets. We need to show that [bigcap C_i]_p = bigcap C_i. Since the other inclusion is trivial, we need to show that [bigcap C_i]_p subset bigcap C_i. Assume, in contrary that there exists x in [bigcap C_i]_p not in bigcap C_i. Let i be such that x not in C_i. Since C_i is closed, x not in [C_i]_p. But now [bigcap C_i]_p subset [C_i]_p, and x not in [bigcap C_i]_p. A contradiction.87.108.22.204 (talk) 12:14, 9 January 2012 (UTC)

I am not sure I know what you're getting at. The article defines a set A to be closed if and only if $A=[A]_{p}$ and it defines it to be open if and only if $X\setminus A$ is closed.

To demonstrate a topology, I need to show that if U, V are open by the above definition, then $U\cup V$ and $U\cap V$ are also open. This is more or less straightforward, (I did verify this, as I was concerned, and can reproduce the proof here if it helps). The one part that I did not do is to verify that, even under an infinite bunch of unions, the result is still an open set. I am not sure I know how to do that. Is that the objection?

FWIW, I cribbed most of the article while sitting with the given reference in my lap; that book makes the claim that the collection of all open sets generated by the praclosure operator is a topology. That this collides with the article on pretopological space is what concerned me. linas 02:58, 28 November 2006 (UTC)

Proof

I'm going to write this down before chucking my scrap of paper in the trash, because I'd be in pain if I had to re-invent this. If U, V are open, then proving that $U\cap V$ is open is easy (so I won't write it here). The proof that $U\cup V$ is open is a little bit harder. It depends on the following lemmas:

Lemma 1: One has $[A\cap B]\subseteq [A]\cap [B]$ which holds for any A, B.

Proof: Note that

$[A]=[A]\cup [A\cap B]$

and

$[B]=[B]\cup [A\cap B]$

Right? Yes, since one has

${\begin{aligned}\;[A]\cup [A\cap B]&=[A\cup (A\cap B)]\\&=[(A\cup A)\cap (A\cup B)]\\&=[A\cap (A\cup B)]\\&=[A]\end{aligned}}$

Soo ... Combining and distributing:

${\begin{aligned}\;[A]\cap [B]&=([A]\cup [A\cap B])\cap ([B]\cup [A\cap B])\\&=([A]\cap [B])\cup ([A]\cap [A\cap B])\cup ([B]\cap [A\cap B])\cup ([A\cap B]\cap [A\cap B])\\&=([A]\cap [B])\cup [A\cap B]\end{aligned}}$

and so, for any A, B, one has

$[A\cap B]\subseteq [A]\cap [B]$

I am often error prone, but I believe this is QED.

---

Well, I'm paranoid. Whenever you challenge me, you end up being right. So one smaller lemma, just in case:

Lemma 2: If A and B are closed (i.e. $[A]=A$ and $[B]=B$) then $A\cap B=[A\cap B]$

Proof: Axiomatically, one has

$A\cap B\subseteq [A\cap B]$

and by Lemma 1, one has

$[A\cap B]\subseteq [A]\cap [B]$

but since A and B are closed,

$[A]\cap [B]=A\cap B$

so

$A\cap B\subseteq [A\cap B]\subseteq A\cap B$

and by reflexivity, equality must hold. QED.

linas 03:39, 28 November 2006 (UTC)

Well, I conceded closure under intersections ... Charles Matthews 15:31, 13 April 2007 (UTC)

OK, this then begs the question of "what the heck is a pretopological space?", since, from what I can tell, the above argument shows that pretopological spaces have topologies, and are therefore topological spaces. So what property makes them "pre-"? Perhaps I'm being dense. linas 03:21, 7 June 2007 (UTC)

You should add another lemma, maybe: that $([A]\cap [A\cap B])=[A\cap B]$. It's really straightforward using monotonicity instead of the binary union axiom. Perhaps the article should include that as well.

What I seem to see from the discussion above is that Charles Matthews is unconvinced the collection of "open" sets given by the pretopology is stable under infinite union, in other words that your binary proof doesn't extend to the case of arbitrary infinite intersections of closed sets.

From what I've read about "convergence spaces," there is a natural induced topology, but the convergence structure induced from that topology is different from the original convergence. Could something similar be going on here? Perhaps the "induced topology" is meant to be generated by taking the pretopology given by the praclosure as a base.

Finally, any words on etymology? I still think "preclosure" or even "proclosure" or "paraclosure" makes more sense. —vivacissamamente 05:45, 7 June 2007 (UTC)

Pre-

Why is this concept called a "praclosure" rather than a "preclosure"? It would seem the later term goes along nicely with pretopological spaces, while the former doesn't initially make sense. Where does it come from? —vivacissamamente 23:13, 27 May 2007 (UTC)

Wikipedia seems to be essentially the only source on the web for "praclosure", and also for prametric space. Both articles were started by User:Linas, who may not have had his mathematical education in an English-speaking country. In German we use "prä" rather than "pre". Many speakers of English would render this incorrectly as "pra" ("prae" would be correct, as the two dots were a mediaeval abbreviation for an omitted e). Perhaps something similar happened here.

I have found a good reference (Banaschewski) for "preclosure" with the meaning defined in this article [1]. I will try to fix this. --Hans Adler 13:11, 15 November 2007 (UTC)