Definition talk:Limit Point/Filter

As reported by private email to the ProofWiki admin (corrected for grammar):

"The first definition is nonsense: the full set $S$ is an element of the filter $\mathcal F$, because filters are closed under superset. Therefore, its complement relative to $S$ is the empty set. If the empty set is one of the sets of the intersection, the intersection must be empty. Therefore, there is NO element which is a limit point of a filter.

"The second definition is different, not equivalent, but it relies on the topological notion of a neighborhood." -- Scott Engles

This matter will be attended to in due course. --prime mover 22:35, 13 July 2012 (UTC)

Although the email may not have been the friendliest, it is correct. The first definition excludes absolutely everything, and therefore cannot possibly serve any purpose. --Dfeuer (talk) 18:26, 9 February 2013 (UTC)

Finally got round to sorting this out. Haven't a clue where I got that first definition from. It was cut from whole cloth at the same time the alternative definition was -- but I can find only that second one in the source work quoted. Deleted, with extreme prejudice. --prime mover (talk) 22:59, 7 January 2015 (UTC)

Limit point of a filter vs filter converges to a point

I've changed the definition and would like to explain why. The source used for the prior definition ("counterexamples in topology") uses the notions "filter F converges to a point x" and "x is a limit point of a filter F" interchangeably. However, in this wiki the latter has a different meaning which I tried to reflect. In particular, a filter $\mathcal F_A = \{F\mid A \subset F \}$ rarely converges (only if the set $A$ is a subset of any neighborhood of any point of $A$), but it always has any point of $A$ as its limit point, according to the usage on this wiki.

I don't have the sources using such definition so I just deleted the old one, disagreeing in notation. l3erdnik (talk) 09:31, 15 May 2019 (EDT)

I have problems with this approach. If the definition given in S&S is "wrong", then we should raise a page (or a series of pages) pointing out where and why it is wrong, with a view to clarifying the position. If it is not "wrong", but instead (as often happens in topology) is the result of differing terminological detail, then we most certainly do not delete it.

Your definition is (as you admit) unsourced, and there is no obvious course of direction to go to investigate the details. If you are particularly knowledgeable about this area of mathematics, and you are able to distinguish between the various definitions, pointing out where the differences lie, then go to it.

Furthermore, iff the definition you provided can be shown to be logically equivalent to the definition given here, then the approach is to add it as a second definition, and a page written to prove that equivalence.

I have reverted the edit, with a view to opening a dialogue on the subject, where you are invited to point out the specific motivation (rather than just the "in this wiki" comment -- where "in this wiki"?) for your change of definition. --prime mover (talk) 16:53, 15 May 2019 (EDT)

I appreciate the openness for dialog.

My motivation was stemming from the fact, that any page that I was able to find on ProofWiki (this, that, etc) that uses the notion of a "limit point of a filter", uses it in the sense "the point is an adherent point of any element of the filter". That makes all the statements correct and the proofs work fine. The only problem that I have is that the definition page itself describes the notion in a different way, that is no longer consistent with the usage of the term in other pages (and happens to be equivalent to another notion). Resolving that inconsistency - directing to the definition inconsistent with the way it was used - was my aim.

Admittedly, it may be a better course of action to leave this definition as is and instead pick another name for the actually used notion and redirect links there, but this is the question of the notation convention of ProofWiki and, I guess, is up to you to decide. l3erdnik (talk) 09:43, 16 May 2019 (EDT)

The problem stems from the facts that:

a) this and that were written by someone who did not actually write a definition of the concept he used.

b) The definition as given here was taken from Steen and Seebach who (as far as I am aware) know what they are doing.

Now I don't want to remove the definition as given here (and we certainly don't just delete the "Sources" section for obvious reasons that you will appreciate), but we obviously do need to include the definition which makes this and that correct and consistent. And, as I say, we need to make sure that the differences between the two definitions are established. --prime mover (talk) 09:58, 16 May 2019 (EDT)

So what would be the solution? Say, Bourbaki in "Elements of mathematics: General topology" calls this relation "the point is a cluster point of a filter". However, this term is already reserved here for a different concept, and the concept needed for the pages I cited doesn't have a term for it over here, it seems.

I mean, I was glad to resolve the contradictions, but I'm not quite willing to hunt through the literature in hopes some book uses a name that is not already reserved here for something else...

Anyone else want to join in? I'm all out. --prime mover (talk) 14:49, 16 May 2019 (EDT)

Personally, I think we should just change the definition here and put prominent Also known as and Also defined as sections. I tend to feel that the terms we use to describe these concepts are less important than the math itself: while it's important to make sure that the concept is established in literature, to make sure we're interpreting it correctly, it doesn't matter that much if we use a different word for it. That source you mentioned that calls it a cluster point should suffice. And really, is finding one obscure source that uses this term (contrary to many others) any different from just making it up? --CircuitCraft (talk) 18:51, 25 March 2024 (UTC)