Talk:Isolated point

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]