Talk:Isolated point

Merge with Acnode

Isolated point and Acnode are terms for the same thing but have their own pages, one of which is a stub. Is there any reason not to merge these? --Pleasantville (talk) 20:45, 25 March 2008 (UTC)

Isolated point is a topological concept and Acnode occurs in the context of algebraic curves. They are the same if you consider the curve as simply a collection of points but usually curves are assumed to have a differential structure as well as a topological one and they can be extended to complex numbers instead of just reals. A more accurate, if somewhat opaque definition of acnode is:

When a curve cuts itself once at the same point, the latter is called a double point, and the curve has two tangents at this point. When the two tangents are distinct, the double poiut is called a crunode or shortly a node; when they are imaginary, the point' is called an acnode or a conjugate point; and when they are coincident, the point is called a spinode or cusp.

(from "An elementary treatise on cubic and quartic curves" By Alfred Barnard Basset)

Perhaps the crunode, acnode and spinode artlces should be merged into a single article.--RDBury (talk) 08:53, 31 August 2009 (UTC)

It's been a month with no movement on this proposal so I'm removing the tags.--RDBury (talk) 11:19, 4 October 2009 (UTC)

Set vs topological space in lead

It seems like "isolated point" is a topological concept, not a set-theoretic concept, because it depends on the notion of a neighborhood. If so, then the first sentence should read

In topology, a branch of mathematics, a point x of a topological space S is called an isolated point of S if there exists a neighborhood of x not containing other points of S.

and other mentions of set in the lead may need changing, too. Mark viking (talk) 22:25, 28 January 2013 (UTC)

Image with Vietnamese text

At the moment, the image, file:Điểm cô lập-Isolated point.jpg, has embedded Vietnamese text. Nothing against the Vietnamese language, but I think it's a bit distracting to have it show up unmotivated in a mathematics diagram in the English WP. Anyone feel like making an image without it? --Trovatore (talk) 00:20, 13 June 2013 (UTC)

I agree -- this is annoying. Looks like it's been 3 years too. Simplyianm (talk) 19:30, 4 October 2016 (UTC)

Reversion of counterexample

The following text that I added to the "A Counter-intuitive Example" section has been removed:

Another set F with the same property can be obtained by choosing one point (e.g. the center point) from each component of the complement of the Cantor set in [0,1]. Each point of this set will be isolated, but the closure of F is the union of F with the Cantor set, which is uncountable.

I suppose the phrasing may be unclear, but I don't see any problem with this example, and I tend to think that's it's simpler than the other example given in this section. Can we put it back in? Jim.belk (talk) 04:25, 26 September 2016 (UTC)

Sure, let's see where my misunderstanding lies: As I understand it, you're defining F to be (for example) the set of all fractions with denominator 2^n (except 0 and 1), right? —Nightstallion 07:03, 26 September 2016 (UTC)

No, wait, that's wrong already. But for that very reason that's the first part that would need to be clearer IMHO. ;) —Nightstallion 09:56, 26 September 2016 (UTC)

The Cantor set is obtained by starting with the interval [0,1] and removing a sequence of open subintervals. The set F is obtained by choosing exactly one point from each of these removed subintervals.

Note that the first few of these subintervals are (1/3, 2/3), (1/9, 2/9), (7/9, 8/9), (1/27, 2/27), .... The set that you mention (namely the dyadic rationals in [0,1]) does not have the specified form. Indeed, 1/4 is actually an element of the Cantor set.

I don't really see what the ambiguity is in the text that I wrote, but I'd be happy to write a new version with the hope of making the example clearer. Jim.belk (talk) 13:27, 26 September 2016 (UTC)

Please do. As it is now, it's not clear to me how the entire Cantor set is in the closure of F. Thanks! —Nightstallion 07:17, 27 September 2016 (UTC)

I've added a new version. You're welcome to edit it for clarity if you think it's still unclear. Jim.belk (talk) 11:52, 27 September 2016 (UTC)

bad display of article upon exit from Talk

If I click on Talk, then log in, then click on Article, the article does not display correctly. (The diagram hogs the page.) Kontribuanto (talk) 11:19, 23 February 2024 (UTC)

wrong logic of alternative definition

The following is incorrectly-worded: "Another equivalent formulation is: an element x of S is an isolated point of S if and only if it is not a limit point of S." The correct wording would be: "Another equivalent formulation is: an element x of X is an isolated point of S if and only if it is in S and is not a limit point of S." (That is, you have to let x be 'freely-floating' throughout the space X to begin with. For all the reader knows initially, some point in X not in S might be an isolated point of X, and this has to be allowed for, and then only later excluded.) Kontribuanto (talk) 11:21, 23 February 2024 (UTC)