Talk:Locally connected space

Some comments on the article

1) In the lead: "surprisingly" is bad prose in mathematical exposition. How do you know whether or not the reader will be surprised? (I am not surprised.)

From experience, I have noticed that quite a few people immediately come to the conclusion that connectedness implies local connectedness. You obviously know that this is not true but some people think that it is. I don't find it surprising either (it is surprising (in a sense) compared to other topological properties).

Topology Expert (talk) 09:23, 7 October 2008 (UTC)

2) You do not give any of the history of local connectedness or any indication why local connectedness is important. Think about this -- why is it important? One idea is the relationship to local simple connectedness and the existence of universal covering spaces.

I will try to do this. Basically (from what I know), local connectedness is an important hypothesis and is assumed in many important theorems (for instance the existence of a universal covering space as you mentioned).

Topology Expert (talk) 09:23, 7 October 2008 (UTC)

3) There are no inline citations in the article. This makes verifiability difficult.

These theorems are quite standard but for WP:V I gave Munkres as a primary reference. Most of the theorems are either proven in Munkres or given as easy exercises. Is there a way to let the reader know that Munkres is the primary reference?

Topology Expert (talk) 09:23, 7 October 2008 (UTC)

4) Why is it necessary to reference linear continuum to explain why the real numbers are connected and locally connected?

What I think that I had in mind was that an order topology is connected iff it is a linear continuum. In particular, every linear continuum is also locally connected. This generalizes to aribitrary order topologies (and not just for R).

Topology Expert (talk) 10:57, 7 October 2008 (UTC)

5) The phrase "elementary sets" links to an article on Lebesgue measure which does not say anything about elementary sets. Again, why is this necessary?

I agree. It is not necessary (I linked it to Lebesgue measure because some contructions of it utilize elementary sets).

Topology Expert (talk) 10:57, 7 October 2008 (UTC)

6) The way you recall the definition of the topologist's sine curve is rather clumsy. In my opinion, better would be to link to the definition and reproduce the picture.

Thanks for the advice. That is exactly what I did (would it be useful to mention that it is the continuous image of a locally connected space that is not locally connected?).

Topology Expert (talk) 10:57, 7 October 2008 (UTC)

7) In local path connectedness: "Similar examples to the previous ones, show that" The comma should be removed.

Done.

Topology Expert (talk) 10:57, 7 October 2008 (UTC)

8) Example 1 would be better phrased as: "Since any path connected space is connected, it follows that any locally path connected space is locally connected."

Changed that.

Topology Expert (talk) 10:57, 7 October 2008 (UTC)

9) "this is trivial" is unencyclopedic.

Done.

Topology Expert (talk) 10:57, 7 October 2008 (UTC)

10) You seem not to mention the fact that a connected, locally path connected space is path connected, which is one of the most useful applications of local path connectedness.

It is actually mentioned in the 'Theorems' section (theorem 4).

Topology Expert (talk) 10:57, 7 October 2008 (UTC)

11) weakly locally connected: Is this concept really notable enough to be included here? Why? (Note: all caps for NEIGHBOURHOOD is unencylopedic.)

Yes, since every weakly locally connected space is locally connected, this is not a very interesting property. But I chose to include it anyway since it is referred to in a couple of books (such as Munkres). Also see: [1].

Topology Expert (talk) 11:04, 7 October 2008 (UTC)

12) Components: "Intuitively, the component are the largest possible connected subsets of X..." Why intuitively? That is exactly what they are -- maximal connected subsets of X.

Yes, your wording is more appropriate.

Topology Expert (talk) 11:13, 7 October 2008 (UTC)

13) Explanations of facts should not be put in parentheses unless they are very brief. Especially, avoid nested parentheses.

14) Why are these proofs necessary?

The proofs serve as justification of the material (for WP:V). WP:OR (if there is any question of WP:OR!) is not a problem; these theorems are standard and Munkres handles them anyway.

I will look into this and see whether I can omit/shorten some of the proofs.

Topology Expert (talk) 11:13, 7 October 2008 (UTC)

15) I find the following alternate treatment of connected components simpler and cleaner:

Step 1: Define a connected component to be a maximal connected subset of X. Since the closure of a connected set is connected, connected components are always closed.

Step 2: Since the union of a chain of connected subsets is connected (maybe this sohuld be remarked upon in the article on connectedness), Zorn's Lemma applies to show that each point x lies in at least one connected component.

Step 3: Since the union of two connected subsets with nonempty intersection is connected, the connected components are disjoint. Therefore they partition X.

Maybe I could add this (nice) treatment after I have polished up the main part of the article.

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

16) The first three examples in the section on components seem to be overly similar to the previous examples.

17) "We will prove later that the path components and the connected components are equal when X is locally path connected." Unencyclopedic tone. Also, this fact should have come earlier, in the discussion of local path connectedness.

I am working on fixing such sentences.

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

18) Example 7: Simpler and more useful is to state that a continuous map from a connected space to a totally disconnected space is constant. Similarly a continuous map from a locally connected space to a totally disconnected space is locally constant, and this can be one of the motivations for studying the concept of local connectedness.

Good idea. I have included this generalization in the article. Could you please tell me how (in particular) this property can serve as a motivation for the study of local connectedness?

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

19) Section on quasicomponents: this section could be about half as long with a tightened up exposition.

I will try to do this. Could you please give me your advice on this: would it be appropriate to delete examples that basically state the statement of a particular theorem? I would think yes, but maybe in such cases there should just be a link to the particular theorem (in the article).

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

20) Section on theorems: this is poor organization. Statements of theorems should be incoroporated into the main text of the article. Each theorem should be given an inline citation. Proofs can be included if they are brief and enlighening, but need not be. Obviously there is no merit in writing "Proof: The proof is similar to theorem 1 and is omitted."

Removed the proof of theorem 2. The citation for (most) of the theorems in Munkres. Could you please tell me how I can specifically let the reader know that this is the main citation?

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

21) Applications of the theorems: the rational numbers are obviously not locally connected, since they are totally disconnected and nondiscrete.

At the time, I thought that it might be useful to include a few trivial examples. But I agree with you so I deleted two trivial examples in this section.

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

22) Again, "elementary subset" of R is not defined.

I removed this example since it is not necessary (as you mentioned)

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

23) "See the infinite broom (or the broom space). This is an example of a space which is weakly locally connected at a particular point, but not locally connected at that point. The reader is directed to Wolfram MathWorld for a definition of the broom space. See also the section (in this article) on ‘weakly locally connected space’." If the concept of a broom space is notable, it should get its own wikipedia article. Mathworld is not a primary source.

At the time that I wrote this article, I was thinking of creating another article on the broom space. But since such an article should have a lot of images (and currently there are no images of the broom space in Wikipeda), I thought that I would do this later.

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

24) None of the three Mathworld references provide any additional insight on the article.

I put them for WP:V (so there is an additional source for the definitions).

Topology Expert (talk) 13:45, 9 October 2008 (UTC)

25) The relevance of the second and third external links is obscure.

Plclark (talk) 19:59, 6 October 2008 (UTC)

Re 1), agree that "surprisingly" is not particularly encyclopedic. However it is unusual when contrasted with other localized properties, e.g. locally compact, locally path connected. Paul August ☎ 21:09, 6 October 2008 (UTC)

I don't understand. If P is any topological property, then "locally P" means "Each point has a neighborhood base consisting of sets having property P", and then it certainly need not be the case that P implies locally P. This is the case for local compactness (of non-Hausdorff spaces) and local path connectedness (as the article already makes clear). Plclark (talk) 00:02, 7 October 2008 (UTC)

Yes, certainly it is not the case that P implies locally P, as many examples show. But P and locally P usually are related. For example, as you point out, compact Hausdorff spaces are locally compact, which follows from the fact that a Hausdorff space is locally compact iff each point has a compact neighborhood; a connected locally path connected space is path connected; a locally metrizable Hausdorff space is metrizable iff it is paracompact. The point is there are apparently no such theorems relating connectedness and locally connectedness. Paul August ☎ 18:17, 7 October 2008 (UTC)

Thanks Plclark for your effort in this and I will definitely try to follow your suggestions. I have to go now but I will respond to all your comments (and fix up some of the necessary material) by tomorrow.

Topology Expert (talk) 09:22, 7 October 2008 (UTC)

Products and quotients

This article should include the results on products and quotients. Paul August ☎ 21:30, 6 October 2008 (UTC)

Sounds good. What results in particular do you have in mind? Plclark (talk) 08:36, 7 October 2008 (UTC)

The immediate results are:

• A product of locally connected, connected spaces is locally connected

• A finite product of locally connected spaces is locally connected

Did you want to include results regarding the local connectedness of box products?

Topology Expert (talk) 10:39, 7 October 2008 (UTC)

Products:

• A nonempty product space is locally connected iff each factor is locally connected and all but finitely many factors are connected.

Quotients:

• Every quotient of a locally connected space is locally connected. This implies, for example, that continuous open images and continuous closed images of locally connected spaces are locally connected.

Paul August ☎ 17:03, 7 October 2008 (UTC)

Some edits made

I made a first round of edits. Aside from removing "surprisingly" from the lead, all of my changes were concentrated on Section 1, examples. This section is now shorter, with a tighter exposition, but yet covers more material than the previous version. The rest of the article deserves a similar treatment, but I want to give TopologyExpert a chance to respond to my comments. Plclark (talk) 03:06, 7 October 2008 (UTC)

Updated comments

The article is now coming along, I think. Here are some issues to think about:

(i) As I pointed out here, there is actually no difference between the definitions given for "locally connected at p" and "weakly locally connected at p". It seems though that what is intended for "locally connected at p" is the existence of a neighborhood base of connected open subsets. As discussed here, I prefer the given definition which does not require the base to be open on general grounds of terminological consistency. But either way, the example of the broom space which shows that the two definitions can disagree at a single point together with the comment that they are the same when required at all points of any given space should stay in the article, and probably be given more prominence towards the lead.

I think the definition requiring U to be open should be placed instead of the one you adopted. From personal experience, I don't think the problem is minor. I found the article after experiencing problems with the definitions in Munkres (which are actually the same as the ones I found here). It took me a while to figure out (after reading the corresponding articles in MathWorld) what each of the authors meant by "locally connected at x".

Presently, I don't understand why you prefer not requiring U to be open "on general grounds of terminological consistency". You associate the same definition to two different terms with only a few paragraphs between, and link them by the claim that the latter is weaker. This, together with the fact that the wordings used in each case are also different, may lead even a passer-by to spend time trying to figure out a difference in meaning. Marcosaedro (talk) 05:49, 14 October 2008 (UTC)

(ii) Regarding proofs of theorems, the question I want editors to think about it: what is the motivation for providing the proof of a theorem rather than giving a citation to the proof in a standard reference (e.g. Munkres)? I am not saying that no proofs should appear -- certainly my edits contain many brief proofs -- but rather that one should think what is the value added for any given proof.

(iii) There is an issue here that this article contains a lot of information on components (and path components and quasi-components). This is certainly material that should appear somewhere on wikipedia, but it is less clear that it should appear here rather than, say, in the article on connectedness. Plclark (talk) 17:29, 7 October 2008 (UTC)

Introduction

I changed the last sentance of the first paragraph. It said that locally connected and connected are independent properties. This is misleading since every connected space is automatically locally connected, and so there is an implication: if connected then locally connected. It follows that they can't be independent. To see this just take the neighbourhood U in the introduction to be the whole space X. The converse is clearly false, i.e. not every locally connected space is connected. To see this let X to be the set of real x such that |x| > 1.  Δεκλαν Δαφισ   (talk)  20:31, 24 November 2008 (UTC)

This is false. U=X might not be contained in V. The topologist's sine curve is a counterexample to your claim that connected implies locally connected. Geometry guy 23:26, 24 November 2008 (UTC)

Ok, it's not locally connected at (0,0) since if you take V to be the subset of the topologist's sine curve contained in the open ball of radius half, no neighbourhood of (0,0) contained in V is going to be connected. Cool, but maybe we should explain this a bit more in article? Ben (talk) 23:49, 24 November 2008 (UTC)

I agree. Maybe the Comb space is a better counterexample, as it is obviously connected and path connected (by the x-axis) yet clearly not locally connected (at nonzero points on the y-axis). (Mathworld has a picture.) Geometry guy 00:08, 25 November 2008 (UTC)

Ahh their picture is great. I agree that the comb space counterexample is a better choice, since it's construction is so straight forward. It would be good to leave a mention of the top sine curve though, something like "and similarly, the top sine curve is connected but not locally connected"? Ben (talk) 00:27, 25 November 2008 (UTC)

The problem with Declan's argument is that every neighbourhood of a point must contain a connected neighbourhood of that point (not at least one neighbourhood) so carefully look at the logic before concluding something false (that is why logic is so important in mathematics). Anyway, I don't understand why the comb space is 'easier' than the topologist's sine curve; in fact the construction of the topologist's sine curve is more likely to be straightforward. Best to keep both examples.

Topology Expert (talk) 20:51, 7 December 2008 (UTC)

New revision

I made the first steps of a substantial revision, somewhat along my comments above. So far I did two different things:

(i) I added an actual introduction, in which I attempted to provide context and analyze the importance of these concepts. If it stands up to scrutiny, some references will need to be added.

(ii) I started to reorganize and pare down the material in the body of the article (which was, and is still, too long for the amount of information it conveys). I decided to concetrate on local connectendess versus weak local connectedness versus local path connectedness and put all three of these definitions together in the first section. Then I merged various example sections and removed redundant material.

Probably the next step is to incorporate the theorems into the body of the article, removing and/or shortening proofs as seems appropriate. Also the material on quasicomponents seems peripheral; perhaps it should be moved to a separate article. Plclark (talk) 06:47, 26 November 2008 (UTC)

I have now completed a fairly substantial revision, which has significantly shortened the article while maintaining almost all of the same content (and adding a little bit as well). I also put in many inline citations to results, mostly from Willard's book, which has a very complete treatment of this material. (My other bedside topology reference, Kelley, says almost nothing about connected spaces.) This is a natural stopping point for me; I would appreciate further input on where the article should go from here. Plclark (talk) 01:54, 27 November 2008 (UTC)

You have done a super job. It is now looking much more like a Wikipedia article should. I will comment again at the WP:Peer review/Locally connected space/archive1 peer review if I have time this weekend. Geometry guy 10:05, 27 November 2008 (UTC)

Altered reference

I have removed <ref>Willard, Chapter 8, example 27.15, p. 201</ref> from the local connectedness vs weak local connectedness section and replaced it with a reference to Steen & Seebach. The space in question is defined more clearly in the latter, and in any case I do not think either is the source of the example. 194.66.154.144 (talk) 16:12, 8 November 2011 (UTC)

"Broom space" example not given...

The text refers to an example, "broom space" that is not given in the list of examples. I imagine a "broom space" is rather linke a "comb space", but I don't think the comb example fits the text reference exactly. I don't know enough topology to make the fix myself, but the omission is glaring. Baon (talk) 18:29, 17 October 2012 (UTC)

Quasicomponents Example 1

Example 1 is unclear: "... a countable set, X, with the discrete topology along with two points a and b such that any neighbourhood of a either contains b or all but finitely many points of X, and ..." This seems to imply that the set {a,b} is open; but since $\{a\}\cup X$ is also open, their intersection {a} must also be open. So the space is discrete. This is not what is intended. Chrystomath2 (talk) 08:06, 4 March 2014 (UTC)

Properties, number 3

The affirmation is false, I tried several hours to prove it (for local reasons) but I found a counterexample, take X the sequence 1/n together with the limit point as a subspace of the reals. Then the sequence is locally connected since each singleton is a local basis of connected open sets (the sequence has a discrete subspace topology in X), and the singleton of the limit point is also a locally connected space. X is the disjoint union but is not locally connected. Too bad I trusted wiki for this one, pretty serious mistake here in the page. 148.240.30.230 (talk) 21:24, 14 March 2015 (UTC). Mhmmm I didn't consider it was the disjoint union topology, not the disjoint union as sets. 148.240.30.230 (talk) 21:38, 14 March 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Locally connected space/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs references overall, and specific citations for each of the theorems and examples. Consider moving the proofs to a separate subpage or out to Wikibooks. silly rabbit (talk) 14:02, 11 May 2008 (UTC)

Last edited at 21:58, 6 October 2008 (UTC). Substituted at 02:18, 5 May 2016 (UTC)

Weakly locally path connected

@Mgkrupa: and others: Do you think it would be useful to add a definition for weakly locally path connected = path connected im kleinen at a point and for the whole space? (with the obvious definition to parallel weakly locally connected). A space is locally path connected iff it is weakly locally path connected (same proof as locally connected iff weakly locally connected, no need to give such a proof if we decide to add the notion). The extended broom space example 119.4 in Steen & Seebach space is weakly locally path connected at the "apex" point, but not locally path connected at that point.

See also https://math.stackexchange.com/questions/3624004. A google search for weakly locally path connected also gives a few hits, for example some article by Saharon Shelah in Proceedings of the AMS, and a recent article in the arkiv (https://arxiv.org/pdf/1002.3583.pdf). But the notion does not seem to be used that much (as it is really the same as without weakly when taken over the whole space). Still, could be worth a mention?

Note also that different topology texts define things differently. For example Willard and Engelking both defines locally path connected as what would be weakly locally path connected. (I am not suggesting to change the existing Wikipedia definition.) PatrickR2 (talk) 07:04, 16 December 2021 (UTC)

Yes, I think that it should be mentioned that different authors define "locally path connected" in different, but equivalent, ways. Because of how technical the field is, mentioning it might help a reader learning topology (e.g. if they notice that this article's definition of "locally path connected" doesn't agree with that of their book). And because as you mentioned, this information can be summarized in just a few sentences. On a tangentially related note, I think that the entire "Definitions" section can be made much shorter since a lot of its text is repeated. Mgkrupa 07:44, 16 December 2021 (UTC)

"Locally path-connected" listed at Redirects for discussion

The redirect Locally path-connected has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 November 4 § Locally path-connected until a consensus is reached. 1234qwer1234qwer4 19:20, 4 November 2023 (UTC)