Talk:Locally connected space

message: 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)
thread_title:  Some edits made 

message: 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)
thread_title:  Updated comments 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Introduction 

message: 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)
replies: text: 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)
thread_title: New revision

message: I have removed Willard, Chapter 8, example 27.15, p. 201 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)
thread_title:  Altered reference 

message: 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)
thread_title:  "Broom space" example not given... 

message: 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)
thread_title:  Quasicomponents Example 1 

message: 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)
thread_title:  Properties, number 3 

message: Substituted at 02:18, 5 May 2016 (UTC)
thread_title: Assessment comment

message: 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)
replies: text: 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)
thread_title:  Weakly locally path connected 

message: 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  until a consensus is reached.  1234qwer1234qwer4 19:20, 4 November 2023 (UTC)
thread_title:  "[[:Locally path-connected]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]