Talk:Glossary of topology

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What is a glossary? It's a specialized type of annotated (stand-alone) list article, where the annotations are descriptions of the terms listed. Glossaries serve the primary functions of lists as well as present definitions to assist topic identification, link selection, and browsing. List structuring and annotation is covered in WP:LISTS, and glossary formatting is covered at Wikipedia:Manual of Style (glossaries).
 


Untitled[edit]

I think it is a good idea to have pages like this, where many short descriptions of related terms are put on a single page, instead of on separate pages. It looks more substantive and makes for better readability. -Joel Schlosberg


I moved this from the subject page:

For instance, {x: 0<|x|<1} is a punctured neighbourhood of 0 in the real line, because {x : 0 ≤ |x| < 1} is a neighbourhood of 0.
Is this example correct? The explanation of open set seems to mean that the given neighbourhood of 0 is not open, but the explanation of Neighbourhood from above states that the neighbourhood of a point contains an open set containing the 1-point set {p}. But The example cannot contain such an open set because 0 is the boundary.

The given neighborhood of 0 is in fact open. It is the open interval (-1,+1). 0 is not in the boundary. Maybe you missed the absolute value sign. AxelBoldt 17:36 Oct 31, 2002 (UTC)


Is the definition of simply connected correct? I thought the base point had to be fixed for all time. Revolver

It doesn't make any difference (although this is not obvious). --Zundark
Thanks. I asked a couple topologists, who said that it didn't matter, although it's sometimes defined this way, you go on to prove afterward that it doesn't matter. Funny...no one ever told me that before! Revolver

I'm wondering some things --

Just how "comprehensive" should this glossary be? Potentially, there are many definitions that could be included (say, separation axioms, etc.) that aren't here. It seems that the current glossary is evolving into a "general topology" glossary (i.e. there aren't any definitions of manifolds or homotopy groups, etc.) Certainly manifold theory and algebraic topology have enough terms to warrant their own glossaries as well (eventually!) Also, as the number of entries of this glossary increases, it becomes hard to find something, if one comes here from another article, esp. the first part. While the entries are logically arranged, it's not easy to find a particular entry. This would be fixed by an alphabetical ordering, but this would lose the nice logical arrangement. Could there be 2 articles (one arranged logically, the other alphabetically)? or can both arrangements be incorporated into a single article? Also, it seems that as the number of math articles/content grows (the math content is large compared to other areas, not necc. by # of articles, but by length of content, scope of content) there will be a number of "glossaries" in the math area. I don't even know how many of these are already in existence. Given that one could be made for any number of areas (number theory, geometry, analysis, differential equations, logic, etc.) maybe there could be a meta-link to the glossaries, maybe on the general math article? Revolver

Glossaries bad?[edit]

Thank you, thank you, thank you. To all of you who have contributed to this page: you have done more for my understanding of topology than four different books and a whole lot of papers. This page is a miracle -- nothing else like it exists, anywhere (and believe me, I've been looking for a long time!).

The crucial reasons why glossary style works is that it effectively requires each term have a definition which:

  • is one line long
  • uses only general knowledge and other terms on the same page ("closed under definition-lookup")

I actually favor some technical changes to the wikipedia codebase that would push all pages towards including something similar (so you could build automatic glossaries), but until something like that is implemented, I think this is great.

Megacz 07:36, 31 October 2005 (UTC)


I think a lot of subjects on this page could benefit from a longer, more involved explanation with examples. My fear this is:

  • People may be linking directly to the glossary page, and these links would have to be fixed.
  • A lot of writers might feel there's no need to write a more complete page when the term is listed here.

Also, the inability (or maybe I haven't figured out how) to link to the middle of a page makes the glossary somewhat inconvenient for readers coming in via links. Is there anything nice we can do for them? Maybe a compact index near the top?

Deco 02:34, 29 Nov 2003 (UTC)

Regarding the subjects on the page requiring more explanations and examples, I think this is the point. Saying too much would defeat the purpose. As I understand it, the purpose of a glossary page like this is so that people have quick access to definitions without constantly jumping to dozens and dozens of separate pages. I think it's intended for someone to use as a reference while reading other articles. If someone wants to look into a particular definition more, there will probably be a link to the appropriate article in the definition itself, (or if not, this can be easily added). It's true that many of the definitions in this glossary could have their own page (and many already do have substantial articles), but I don't see why having them listed here would cause people to think a more complete page is not needed. If anything, it might give someone an idea to write more (in essence, they might have the same thought you did).

The other problems you point out are definitely problems. I'm not sure what to say on, e.g. finding particular defs on a page with dozens. Either they can be all alphabetized somewhere, or else the user would need to be aware of the organisation of the defs. Making a compact index (i.e. further breakdowns into categories) might be nice, but it might not help someone who isn't already familiar with the general classification of defs anyway.

As far as fixing links, maybe this is something we should think about, what the purpose of a glossary like this is. Within an article, when you use a term, but don't want to explain it, should you link to a page with just a definition, or an entire article? Or should every page using general topological terms have a short statement somewhere giving a link to the glossary? I know that a page like this is very useful to me, if I keep a separate window open to it, when reading other articles. This could be true in a lot of subjects. It seems that it should be most useful to people exploring articles on their own, but who know enough to learn definitions on their own as well. I think the potential audience for this use is enough to justify it. How it might all turn out in the end, I don't know. Revolver

Two other points I thought of:

  • There are some defs that will probably never justify their own articles (e.g. "subcover", "refinement", etc.) since they are either used in service of other major concepts or just aren't that interesting.
  • Another major purpose of a glossary like this is to provide a standardised set of definitions for wikipedia itself. This is especially important in mathematics, when defs may not always be standard. Here, people would come not only to learn defs, but to learn what the standard defs are for the wiki, which have to be uniform for every article in the entire wiki. This is important, e.g. for the separation axioms. People need to have places to come to to know what standards are in use. Revolver

I took the liberty of organising part 1, it was just getting out of hand. Feel free to move stuff around if you think I put something in the wrong place, it's just a start. Revolver

Removal of basis[edit]

I strongly disagree with the removal of the term basis from the dictionary, because:

  • The page "basis (topology)" redirects to here.
  • It's the only term I knew it by when I arrived; it took me a long time to find the entry for base and realise it was the same thing.

Could this be added back?

Derrick Coetzee 01:26, 1 Dec 2003 (UTC)

I agree. Although I've always been aware of both terms (base and basis), I've heard basis used at least as much as base, if not more. A quick couple of searches on google confirms it's standard terminology. I've changed it back. Revolver

Alphabetical index[edit]

Okay...I know I was very "bold" (to use the wikipedia suggestion) in throwing out the thematic organisation of the glossary in favor of a single alphabetical listing, but I have several reasons why I think this is a much better solution.

Here is the trade-off, as I (and some other real-life individuals I talked to) see it: you can go with a thematic organisation, and this has the advantage that similarly defined or conceptually similar definitions are listed in proximity to each other. So, e.g. you can "view" all the various compactness definitions in a single place, with your eye, and also see how the definitions interact thematically. Another advantage is that the definitions tend to increase in terms of "nested-ness", i.e. they read in roughly the order you would get them in a textbook, so all the terms needed are listed previously. This also has some drawbacks. Despite these advantages, I think alphabetic is better, and here are several reasons I have:

  • Easy to find definitions -- Definitions and terms are quickly and easily located, no matter how familiar or unfamiliar they are to the reader. Compare to thematic organisation; with thematic organisation, if the reader is somewhat unfamiliar with a particular definition or term, then he/she may have a very difficult time locating something. Which section to look in? Often, a scan of the entire glossary would be needed. This is both frustrating and time-wasting.
  • Non-linear logical dependence of definitions -- under thematic organisation, it was very tempting to try to only define something if it was in terms of previous definitions; I think this is a mistake, for reasons I'll go into later. But I ran into this problem with "connected component"; it should obviously go in part 1, since it's not a type of space, yet to define it requires the definition of "connected", which isn't until part 2. So, a linear ordering of the definitions is probably impossible in any case.
  • Alternative terminology/spellings are easily accomodated -- if I say "Kolmogorov" and you say "T0", and I don't know what T0 is, I might have trouble finding it, given that I don't even know the name of what it is I'm supposed to be looking for. With alphabetical listing, all alternative terminology can be listed with "links" to a main entry. Thus, no one is left sorting through the entire glossary to find their definition. (I believe this has already happened in the case of base/basis for some people.)
  • Addition of entries is easily accomodated -- entries can easily be added without making the glossary much more difficult to use. This can't be said for the thematic organisation.
  • What is the raison d'etre? -- The thematic organisation would make more sense if the glossary's purpose was to teach someone, but I don't think that's the purpose. The articles themselves should teach; the glossary is really just a reference for people to quickly look up terms when they don't know (or can't remember) a particular definition. The articles are the main expository meat, I doubt anyone would learn anything directly simply by reading the glossary from top to bottom, without going to the articles themselves. An alphabetic listing makes sense seeing the glossary as a reference.

I already find the alphabetical listing easier to use and visually easier on the eyes than it was before. But I'm interested what others think.

Revolver


As long as one person is making bold changes (which I admit I agree with), I suggest a change bolder still: the elimination of the glossary page in its entirety. Notice that:

  • We already have a quick way to look up a term: using the Go box at the top of the page. Terms within a page should be able to merely be clicked on.
  • We already have a quick way to send one term to another: redirect pages.

We're reinventing the wheel on a local scale. While some like to have a little collection of short definitions of many related concepts, I would argue that if this exists at all it should be generated, not written by contributors. Unfortunately the Wikipedia lacks this capacity. For lack of this, I think the glossary is harmless to have around, but I strongly believe: nothing should link here, and nothing should redirect here, at least once all the proper articles have been created, even for the simple terms. People need a way to look them up more directly.

I would advocate this as a general policy for glossaries on the Wikipedia.

Deco 05:00, 5 Dec 2003 (UTC)


Some general comments:

The original reason for the topology glossary was simply that the topology article was a mess, and I wanted to sort it out. Creating a topology glossary and moving parts of the topology article there seemed like a good idea, so that's what I did. Articles using topological terms could then link to the glossary for definitions. (There were very few articles on individual topological terms at that time.) Of course, I could instead have created a large number of tiny stub articles, but that would certainly have been frowned upon, especially as some of them would have been orphans.

More than two years after the glossary was created, we still don't have individual articles for many of the terms, because the glossary itself has greatly expanded. So the glossary still serves a purpose, as terms defined here but lacking their own article can be redirected here until someone gets around to writing an article. Direct links to the glossary shouldn't usually be used, of course.

As for the organization of the glossary, the separation into two parts was simply something that was convenient at the time, and Revolver is probably right that an alphabetical ordering is now more appropriate.

--Zundark 12:05, 5 Dec 2003 (UTC)


"As long as one person is making bold changes (which I admit I agree with), I suggest a change bolder still: the elimination of the glossary page in its entirety."

I've been thinking about this and I have to admit I've wondering the same thing, or at least I've been wondering more and more what the point of the glossary is (esp. after looking, working on it for a while). In my mind, (which has changed a lot just recently, it seems) there are a number of things working against (having) the glossary:

  • Links can simply be made within articles to other articles.
  • Links to articles contain more information than definitions.
  • No need for a separate glossary page.

There's still some issues that seem problematic for me, though. Working against not having a glossary:

  • It's not always such a simple matter to hit the "go" button to find a definition of something. Currently, the "go" button is based on google, and we all know the reliability/usability of that. If you want the definition of "X", odds are that you might not hit it on the top search match, or even on the top page of search matches. Evidence of this is that sometimes I've actually had to type in the URL itself to get somewhere. I've heard this is a problem with google, so it might be solved with better and better search engines in the future, of course.
  • Even if you do find just the right article you need, it's not always so easy to visually scan to find the definition immediately. There's usually between a line or two to a paragraph of two of introductory remarks before a formal definition (as recommended by the Math Project guidelines), so you sometimes have to look a while. This is more of an inconvenience for people who just want to find the definition quickly, then go right back to their original article, instead of people wanting to read on. Also, several definitions are often found in a single article, since they might make the most sense to define and explain within another general article topic, (although even this seems to change throughout time), and some of these might never get their own article. So, if something is defined near the middle or end of a lengthy article, it can be difficult for someone to find, even if led to that particular article.

Despite this, I wonder if these problems can't be worked around. Some ways to do this:

  • Who cares about stubs?? I know that stubs are considered "bad" on the wikipedia in general, but maybe we should ask, which is better/worse -- a lot of small definition stubs (that could still grow in the future), or an enormous, unweildy glossary?? I know that "wikipedia is not a dictionary", but in mathematics, definitions themselves have a fundamental role that they don't necessarily have in other areas of knowledge (e.g. literary theory or sociology or even chemistry).
  • A practice that seems to be working well, (one which I haven't followed very well myself!) is to try to make your links go to verbatim what you're saying, and then if there's nothing there, do a redirect to the best substitute. Then, if someone comes along later and finds it's a redirect they could start a new article for, they just go in and take out the redirect and start writing. (Example: Stone-Cech compactification -- it's redirected to compactification, although it easily seems possible that at any time someone could go in and take out the redirect and start writing an article for SC-comp. right there.)
  • There could be some standard notation or visual clue that could be used in all math articles to signal definitions (I mean, beyond headings that say "definition" or "formal definition"). The reasons I have for this, again, are that definitions aren't always easy to visually locate, if they're not right at the top, and another reason is that definitions, esp. of secondary topics or concepts, are sometimes hidden within the text itself and not even prefaced by comments or headings. Maybe all definitions could be set off in a box, or have some other way to uniquely set them apart so they're easy to spot.

Revolver

"While some like to have a little collection of short definitions of many related concepts, I would argue that if this exists at all it should be generated, not written by contributors. Unfortunately the Wikipedia lacks this capacity."

There's a wiktionary, isn't there? Can't math definitions go there? I know it's supposed to be a general dictionary, but isn't any specialised branch of knowledge free to use it as well (physics, chemistry, archaeology, art history, etc., etc.??) This would make it easier for people who wanted to find just a definition quickly. Revolver

--- It's not always such a simple matter to hit the "go" button to find a definition of something. Currently, the "go" button is based on google, and we all know the reliability/usability of that.

This isn't exactly true. The only search we have is based on Google, but I visit most articles directly using the Go button, the button labelled go, which only does exact matching.

There's a wiktionary, isn't there?

As you said, definitions in math are quite different, and I think distinct from definitions of the Wiktionary sort. Open up a random dictionary to "homomorphic" to see what I mean.

As for finding definitions in math pages, I think it's reasonable (if unwieldly to fix) for every math page to have a book-style definition which stands out clearly, such as this:

Definition. A set is said to be compact if each open cover contains a finite subcover.

Moreover, I think an article which requires important concepts, such as compact requiring open cover, should give at least an intuitive definition of the concept therein without having to go afield, while providing a link for more detailed information.

Derrick Coetzee 08:12, 6 Dec 2003 (UTC)

As for finding definitions in math pages, I think it's reasonable (if unwieldly to fix) for every math page to have a book-style definition which stands out clearly, such as this:

Definition. A set is said to be compact if each open cover contains a finite subcover.

Yes, this seems like an excellent idea/solution. As for this being "unweildy" to implement or fix, it might not be that much of a problem...there are a lot of people working here. Lots of protocol changes have gone into effect before. And it would be easier to start with it now than later on when there are more articles. Revolver

How about this?:


Definition   Let S = {0,1}. Then T = {{},{1},{0,1}} is a topology on S, and the resulting topological space is called Sierpinski space.


or this?


Definition   Let SL(2,7) denote the group of all 2&times2 matrices of determinant one over the finite field with 7 elements. Then G = PSL(2,7) is defined to be the quotient group SL(2,7) / {I,−I} obtained by identifying I and −I. In this article, we let G denote any group isomorphic to PSL(2,7).


The combination of


along with

Definition   [plus one nbsp]

seems to set things off so its very easy to find, but it doesn't interrupt the flow of most articles. (see Sierpinski space and PSL(2,7)

I have problems with alphabetical lists. They seem to privilege random access to concepts, over being a study guide (that is, taking the chance to group concepts into clusters, impose some sort of reasonable order, and use indentation to show dependencies). I also have a problems with the formal, mathsy style here. It is fine in its place - but that is at most 25% of a good article, sometimes. Discursive is good, in my view.

By the way the

You can define stuff here

style of display seems to me less intrusive, for boxed-off material needing to be displayed.

Charles Matthews 10:22, 6 Dec 2003 (UTC)

The purpose of experimenting with the above changes wasn't to be formal or mathsy, per se. It was just meant to make it easy to find definitions. I know definitions are a really small part of most articles, and certainly wikipedia articles should strive to be anti-Bourbaki, but there are times when you want to know just exactly what is meant by a certain term, precisely, or you just need to know what a term means, and being able to find definitions quickly helps this. Revolver

Well, yes - that probably does apply to me. It would take me a couple of seconds to do a full text search of a long page for Sierpinski. Have to set that against the needs of a more 'average' reader, though.

Charles Matthews 10:40, 6 Dec 2003 (UTC)

I've experimented with a couple pages to see how different things might look, so others can see. In each of these, if you look at the most recent links in the page history, you can see 3 different versions, one without any "setting off", one with lines, one with box. If the idea doesn't catch on, and people wish to change everything back, it's fine. But at least people can compare the difference here. linear combination Sierpinski space PSL(2,7) Baire space Banach-Mazur game Modular group Gamma Revolver

Well, yes - that probably does apply to me. It would take me a couple of seconds to do a full text search of a long page for Sierpinski. Have to set that against the needs of a more 'average' reader, though.

Yes, I know what you mean. It's all a matter of good judgment, probably. On the one hand, I would hate to see accessible articles on prime numbers or basic geometry or calculus get bogged down in endless clutter of definitions getting set off...on the other hand, I doubt that anyone who is needing to look up the different between "paracompact" and "locally compact" or forgot the exact definition of "natural transformation of functors" is going to be shocked at the sight of a definition boxed-off occasionally. Revolver

Glossary word links[edit]

One problem with the glossary is that it's uncertain which entries already have real articles associated with them and which are still functioning as "mini-stubs" until the article is written. To distinguish these better, for any term which has an article associated with it, I link the bold word beginning that definition directly to its article. I haven't finished doing this though, and any help would be appreciated.

I also added more links while I was at it; considering this may be a page that someone consults when encountering a new term which may in turn require more new terms, I think it's deserving of a high number of (meaningful) links.

Deco 07:41, 19 Dec 2003 (UTC)


I'm late to the conversation, but since I've both created a lot of redirects to this page and written a lot of articles on concepts that were once defined only on this page, I thought that I'd give an opinion.

I think that the glossary, if kept short, can serve a purpose if it's linked to by phrases such as "This article uses terms from topology that may be found in the topology glossary.". But one should never create a link using the pipe trick like "open set"! If the article Open set doesn't exist (of course this example does), then it can be redirected here, but all links should go through that redirect.

Also, redirects should go to a more specific page if one exists. Thus if there is no article Perfectly normal space, still that should redirect to Normal space rather than here. That also helps keep the glossary concise; semiregularity is obscure enough that it really doesn't belong here.

So the glossary is a good thing, but it needs to be used wisely. In particular, links should be as specific as possible.

-- Toby Bartels 00:19, 15 Feb 2004 (UTC)


+ Differential topology ??[edit]

Do you think it is ok to add here terms from differential topology?

Tosha 19:32, 6 Mar 2004 (UTC)

It is really for general topology. I think it's better to have a number of separate pages. A very long glossary becomes difficult to use.

Charles Matthews 19:50, 6 Mar 2004 (UTC)

Bursting?[edit]

I realise, esp. with my recent additions, that the glossary is now officially bursting at the seam. I'm not sure how much of a problem this is...it's true, a "very long glossary" becomes difficult to use. But, this is an area with a ton of definitions. Part of the problem is that general topology is the backbone for so many OTHER areas, that even including the most fundamental "helper" definitions creates a long list (semilocally simply connected, homogeneous, etc.) One can argue not to include "helper" defs at all, but then again, one can argue that general topology itself developed as a result of "helper defs" to other fields, (not counting the recent explosion of study of point-set topology for its own sake, that is...) Maybe things like "TG" and "TVS" can be safely removed, since they aren't strictly topological objects (similarly, I don't think "fundamental group" belongs as an entry, since it's an algebraic object). What do people think. Revolver 14:41, 18 Jun 2004 (UTC)

I think the glossary is helpful, and I do not have a problem with the length. I agree that algebraic topology needs its own glossary, and that "fundamental group" does not belong in this one. Rick Norwood 13:41, 31 October 2005 (UTC)

Definition lists[edit]

After my indulgences at mathematical jargon I've become fond of definition lists for pages like this. For one, they just look better, and they take up less space in markup as well. One feature I'd love to see, which I don't see, is being able to link to particular items in the list as one would link to sections of the page; this seems uniquely desirable for this particular list but it's not implemented as far as I know. So: if anyone reads this who knows what's what in the universe of Wikipedia development, could you tell me if this can get implemented? And in the mean time, is it worth putting HTML tags <span id="defn"/> tags in front of each definition, so that when we have a cross reference we link there rather than directly to the main page? This sort of centralizes the control of each term. Ryan Reich 16:43, 29 March 2006 (UTC)


Node, vertex, edge[edit]

Are these terms from topology I learned at school no longer of any use? Hotlorp 15:31, 27 February 2007 (UTC)

2007-02-7 Automated pywikipediabot message[edit]

--CopyToWiktionaryBot 13:32, 7 February 2007 (UTC)

Accessibility[edit]

Currently, the wikitext for the definitions on the glossary page uses empty lines in many places to separate consecutive entries. When rendered as HTML by the wiki engine this will cause the entries to be broken up into sequences of singleton lists. E.g.,

;Accessible: See [[T1 space|'''<math>T_1</math>''']]. 

;Accumulation point: See [[limit point]].

;[[Alexandrov topology]]:  A space ''X'' has the [[Alexandrov topology]] (or is '''finitely generated''') if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed.

(with empty lines between the definitions) will be turned into

<dl>
<dt>Accessible</dt>
<dd>See <a href="/wiki/T1_space" title="T1 space"><b><img class="tex" alt="T_1" src="http://upload.wikimedia.org/math/a/5/7/a5749ec33f2c95fe8c19d702d76d4968.png" /></b></a>.</dd>
</dl>
<dl>
<dt>Accumulation point</dt>
<dd>See <a href="/wiki/Limit_point" title="Limit point">limit point</a>.</dd>
</dl>
<dl>
<dt><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a></dt>
<dd>A space <i>X</i> has the <a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> (or is <b>finitely generated</b>) if arbitrary intersections of open sets in <i>X</i> are open, or equivalently, if arbitrary unions of closed sets are closed.</dd>
</dl>

by the wiki engine, where every definition is wrapped by its own definition list. Without the empty lines, i.e., like this

;Accessible: See [[T1 space|'''<math>T_1</math>''']]. 
;Accumulation point: See [[limit point]].
;[[Alexandrov topology]]:  A space ''X'' has the [[Alexandrov topology]] (or is '''finitely generated''') if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed.

the definition list will be rendered as

<dl>
<dt>Accessible</dt>
<dd>See <a href="/wiki/T1_space" title="T1 space"><b><img class="tex" alt="T_1" src="http://upload.wikimedia.org/math/a/5/7/a5749ec33f2c95fe8c19d702d76d4968.png" /></b></a>.</dd>
<dt>Accumulation point</dt>
<dd>See <a href="/wiki/Limit_point" title="Limit point">limit point</a>.</dd>
<dt><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a></dt>
<dd>A space <i>X</i> has the <a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> (or is <b>finitely generated</b>) if arbitrary intersections of open sets in <i>X</i> are open, or equivalently, if arbitrary unions of closed sets are closed.</dd>
</dl>

with all definitions collected in a single list. Apparantly, people using screen readers can easily navigate between the definitions within a single definition list but not if the list is broken as it is now. See also Wikipedia:Accessibility. I would like to remove the empty lines between the entries if nobody objects. — Tobias Bergemann (talk) 10:43, 20 January 2010 (UTC)

Almost discrete[edit]

There seem to be several definitions in the literature, none of which are obviously the same as that in the article:

  • The only strictly finer topology is the discrete topology
  • Contains no infinite compact subset
  • Exactly one isolated point
  • Hausdorff with exactly one isolated point
  • Every compact subset has an isolated point
  • Punctured neighbourhoods of a point form an ultrrafilter
  • The empty set is the only nowhere dense set
  • Alexandrov and zero-dimensional
  • The set of isolated points is dense

Is there a reliable reference that relates these? Junior Wrangler (talk) 19:28, 11 December 2012 (UTC)