Definition talk:Limit Point/Topology/Set

Some solution to the ugliness of the headings of this page needs to be invented. I am getting more motivated to really start developing a PW extension for MediaWiki, in order to achieve that transclusions are able to adapt heading levels... Now that would be great. --Lord_Farin 09:24, 24 February 2012 (EST)

What ugliness? --prime mover 16:12, 24 February 2012 (EST)

Unnecessary headings which deviate from house style to be able to transclude the page nicely. There might come instances where a workaround like this might not be feasible. I would ideally produce a structure such that one could do the following:

<transclude section='Proof'>Pagename</transclude>

which would transclude the proof (indicated by a custom 'proof' tag) of the page. More options could be added to the 'transclude' tag to allow for a flexible environment making transclusion more consistent and insightful (rather than littered 'onlyinclude' tags in pages, of which one has a hard time finding the use). --Lord_Farin 17:17, 24 February 2012 (EST)

Okay then, that would be of benefit. I admit to having had trouble in the past squeezing the information into the supported structures. --prime mover 17:37, 24 February 2012 (EST)

Incorrect definition?

I seriously doubt that the "definition from sequence" is actually equivalent to the other definitions. For example, let $\omega_1$ denote the smallest uncountable ordinal, i.e., the Hartogs number of $\omega = \left\{{0, 1, 2, \ldots}\right\}$. Consider its successor set $\omega_1^{+} = \omega_1 \cup \left\{{\omega_1}\right\}$ with its order topology. Then $\omega_1 \in \omega_1^{+}$ is a limit point of $\omega_1 \subseteq \omega_1^{+}$, but does not satisfy the "definition from sequence." --abcxyz (talk) 04:01, 12 October 2012 (UTC)

Okay so the context for that one needs to be amended. I'll check what my source work says, I probably misunderstood. The perpetrator of this mistake is to be fired. --prime mover (talk) 06:28, 12 October 2012 (UTC)

In my opinion Definition:Limit Point/Topology/Set/Definition 5 is a duplicate of Definition:Limit of Sequence (Topology). Would that have any implications on how to resolve the 'non-equivalent definition' issue?

My suggestion would be to

(1) No longer include Definition 5 in Definition:Limit Point/Topology/Set

(2) Have Definition:Limit Point/Topology/Set/Definition 5 redirect to Definition:Limit of Sequence (Topology)

(3) Include the link Definition:Limit of Sequence (Topoplogy) in the Section 'Also See' of Definition:Limit Point/Topology/Set

--S.anzengruber (talk) 15:03, 10 July 2015 (UTC)

Sorry for the slight delay. Your suggestions seem sensible, except that I would simply obliterate Definition 5 -- no point in keeping it around. — Lord_Farin (talk) 20:51, 20 July 2015 (UTC)

As has been pointed out Definition:Limit Point/Topology/Set/Definition 5 is not equivalent to the other definitions for a topological space. So it needs to be removed from this page.

While Definition:Limit Point/Topology/Set/Definition 5 makes use of the definition of a Definition:Limit of Sequence (Topology) it is not the same thing.

Definition:Limit Point/Topology/Set/Definition 5 is equivalent to Definition:Limit Point/Metric Space. So Definition:Limit Point/Topology/Set/Definition 5 should be included in the Definition:Limit Point/Metric Space along with the definition that a point is a limit point of A in the metric space iff it is a limit point of A in the induced topological space.

In fact, Definition:Limit Point/Topology/Set/Definition 5 is an equivalent definition of a Limit Point in any First Countable topological space (or Frechet-Urysohn space, but this is precisey a category of topological space where the definition is the definition of a limit point).

--Leigh.Samphier (talk) 10:01, 30 May 2022 (UTC)

Every time I swing past this page I tell myself I need to resolve this, but I never get round to doing so.

If you are of a mindset, feel free to do what needs to be done. --prime mover (talk) 16:36, 30 May 2022 (UTC)

Possible duplication?

Unless I'm missing something, the "definition from open neighborhood" is merely a more symbolic way of writing exactly what is said in the "definition from open set." Could this be an accidental duplication, or is there some purpose for it? --abcxyz (talk) 05:11, 17 November 2012 (UTC)

Good call. Suggest though that somewhere in Definition:Open Neighborhood we emphasise the fact (it *is* true, isn't it?) that "an open neighborhood of (a set) $A$" is exactly the same thing as "an open set which has $A$ as a subset" (and similarly for points). The question remains: do we stick with the definition based on the added layer of technical detail that is "open neighborhood", or retain the definition as based on "open set containing $A$" which is easier to directly assimilate? Actually I think the best approach would be to use "open neighborhood" and then add a "that is" to explain that it means what we already have for Open Set. That would work. --prime mover (talk) 09:07, 17 November 2012 (UTC)

Your suggestion appears sound to me. It's probably best to expose people to definitions with explanation in (particularly) the definitions, so that we can be more confident about using the terminology in proofs (which is bound to happen, especially when and if more people show up to contribute). --Lord_Farin (talk) 09:11, 17 November 2012 (UTC)