Talk:Connected space

Shouldn't a topic like this, where the English word has a meaning, for 99% of the people reading Wikipedia, different from the meaning discussed in the article, live at a page that makes it a little clearer what the topic is? So, instead of "connectedness," how about connectedness (topology) or connectedness (mathematics) or connectedness (math)?

Here's a rule to consider (I should probably put it on the rules page...). If Google does not list any website concerning your topic, on the first page of a search for your proposed topic title (see, e.g., [1]) then you need to make your title more precise.

Soon, there will be a link to the mathematical concept of connectedness on the first Google results page :-)

But seriously: are we ever going to have an article about "connectedness" in the English sense? I guess whe should never say never, but I don't see the need to disambiguate the title before there's another article that it needs to be disambiguated from.

I just searched for "connectedness" in EB: 8 results, and the results number 1, 3, 4, 7 and 8 refer to the mathematical concept. AxelBoldt

By the way, I think it is great that we have people like Axel Boldt writing meaty articles on technical topics. I hope that never changes! By no means is this any sort of criticism of him. --Larry_Sanger

Actually, Zundark wrote all of this; I only contributed by adding some false statements which he fixed. AxelBoldt

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connected if it cannot be divided into two disjoint nonempty open sets

"Two or more" surely? - Khendon

If you have 2 or more, then you take one of them and call it A, and take the union of all the others and call it B. Since the union of open sets is open, anything that satisfies your definition (of "unconnected") will satisfy the article's definition. — Toby 07:59 Sep 18, 2002 (UTC)

Ah, of course. Thanks. - Khendon

I think Toby is wrong here. Because the definition says "...if it IS the union of..." not "...if it can be written as the union of... ". Also, it is much better to use the notation $(X,\tau )$ to denote a topological space in a formal definition instead of $X$ merely. I would say the following definition is better than the current one:

$(X,\tau )$ is a connected topological space if and only if $\tau$ has only two subsets that are both open and closed (clopen) which are the $\emptyset$ and the entire X.

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The present definition is restricted to topological spaces. As far as I can see, this makes non-open subsets like [0,1] in R, or the unit disk in RxR, non-connected. Is this intentional? Am I just missing the connection? rp

You are misunderstanding something, although I'm not sure what. [0,1] and the unit disk are both connected. What makes you think otherwise? --Zundark 15:33 Nov 28, 2002 (UTC)

I think the misunderstanding stems from the fact that rp doesn't realize that any subset of a topological space is a topological space in its own right, with the subspace topology. I've made that clearer in the first paragraph. AxelBoldt 13:57, 2 Aug 2004 (UTC)

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I redirected Path-connected topological space here, since there's more info on that subject here than there. The articles could be separated again, but that would take more work to do the separation properly, and I don't think that it's necessary now. -- Toby 01:34 Apr 24, 2003 (UTC)

merge

I think Connected component's content should be merged herein. I've written as much at Talk:Connected component; please keep discussion there. —msh210 15:44, 5 Dec 2004 (UTC)

hmm

Just found out that G.E. Bredon (ISBN 0-387-97926-3) defines "arcwise connected" exactly as is done here for path-connected. Namely, "a topological space X is said to be "arcwise connected" if for any two points p and q there exists a map $\lambda \colon [0,1]\to X$ with λ(0)=p and λ(1)=q". [p.12] (A map was earlier defined to be a continuous function). Mistake or different uses in different disciplines? Or should I assume path-connected = arcwise connected != arc-connected? \Mike(z) 18:00, 15 May 2005 (UTC)

Different authors may give different meanings to the term arc-connected (or arcwise connected), but I think the one given in the article is the usual one. Part of the problem is that some authors are only really concerned about Hausdorff spaces, so for them there is no real difference between path-connected and arc-connected anyway. --Zundark 13:51, 20 May 2005 (UTC)

Can anyone point me to a book with a proof that path-connected = arc-connected in Hausdorf spaces? --Ucio (talk) 18:00, 22 June 2009 (UTC)

"Formal" definition

It seems to me that the definition in the lead paragraph is better than the one given in the section "Formal definition". Specifically, the lead paragraph refers (by linkage) to Disjoint union (topology), while the "Formal definition" only refers to set-theoretic notions of disjointness and union. There might be other ways of equipping the set-theoretic disjoint union with a topological structure than the canonical one. --Lambiam^Talk 07:56, 16 May 2006 (UTC)

Wish

I'd like to have the ability to have a cross reference button to search for antonyms or opposites of words.

—The preceding unsigned comment was added by 192.35.35.35 (talk • contribs) 01:52, July 24, 2006 (UTC).

Like "What links here"? --Lambiam^Talk 11:04, 24 July 2006 (UTC)

This already exists on our sister project, Wiktionary; see wikt:Wiktionary:Wikisaurus. However, at the time of this writing, it's not very complete, especially if what you're interested in is mathematical terminology. Another good, free resource for antonyms of word is WordNet, but again, it's not terribly useful for mathematical (or otherwise specialist) terminology. —Caesura^(t) 12:45, 6 May 2011 (UTC)

Every path-connected space is connected!?

Is this true?

 Take the real line R with lower limit topology,  then R is not connected, but path connected.

 Take the real line R with finite complement topology, then R is connected, but not path connected.

Jesusonfire 04:56, 15 November 2007 (UTC)

You have those the wrong way around: R with the lower limit topology isn't path-connected, but R with the cofinite topology is. The fact that path-connected spaces are connected follows from the connectedness of [0,1], together with the preservation of connectedness under continuous images. --Zundark 08:18, 15 November 2007 (UTC)

What is a connected set?

Connected set redirects to this article, which, however, does not define the notion. The MathWorld article referred to gives a definition ("cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set"), but I don't think that is right. Any subset S of a topological space X is open in the induced relative topology, because, by definition, X is open, and so S ∩ X = S is an open set of the relative topology (which should be obvious because the construction turns S into a topological space). That makes any set whose cardinality exceeds 1 non-connected. I can make up a definition myself, but does anyone have a citable source for a good definition?  --Lambiam 20:23, 7 January 2008 (UTC)

I see now that this is not a problem; while the set being partitioned is open, the parts it gets partitioned into should use the relative topology for the whole.  --Lambiam 20:52, 7 January 2008 (UTC)

Graphs

In the last paragraph, how can an edge be "homeomorphic" to a line? An edge of a graph is just a pair of vertices, without a topology; it cannot be homeomorphic to anything. What is probably meant is that there is a topological subspace of Euclidean space such that the connected subsets of the graph correspond to (several) connected subsets of the space that include the "vertices". —Preceding unsigned comment added by 81.210.250.14 (talk) 09:27, 26 June 2008 (UTC)

The statement in the article about Euclidean space was not true (for graphs of sufficiently large cardinality) and in any case a red herring. The right way to view graphs as topological spaces is described at topological graph theory. Algebraist 13:35, 21 April 2009 (UTC)

Merge discussion

• Propose that Connected space/Proofs be merged here. This follows a discussion on Wikiproject Mathematics. Itsmejudith (talk) 01:08, 23 February 2009 (UTC)

• Oppose in that the statements proven are already present in this article. The proofs add no value: they are inconsistent in style and content with what is already present here; including them would give undue weight to two relatively elementary and minor technical facts; these facts are standard and uncontroversial, so don't require explicit sourcing; writing our own proofs isn't an acceptable method of sourcing anyway; and including the proofs in a showcase capacity (as opposed to as justifications) seems contrary to the fact that Wikipedia is not a how-to-do-proofs-with-connected-spaces manual. (Summary of my comments at the WPM talk page.) The /Proofs article should go straight to deletion. Ryan Reich (talk) 03:35, 23 February 2009 (UTC)
• Comment I mostly agree with Ryan Reich, except with "writing our own proofs isn't an acceptable method of sourcing anyway" (as long of course as "our own" proofs are trivial variants of proofs available in mainstream mathematic litterature). As concerns "undue weight", we can get around the problem using footnotes or collapsing boxes. Once I have said that, I must agree that these proofs are very trivial and of little value for the article proper, so my conclusion is something like a weak oppose. Since everybody was less lenient than myself towards these poor proofs when they were discussed at Wikipedia talk:WikiProject Mathematics, consensus for deletion is very likely ahead -and I shall certainly not oppose to such conclusion. French Tourist (talk) 07:30, 23 February 2009 (UTC)
• Comment I can't think of a reason why a professional mathematician would read this article (connected space). The only people who read such basic articles (at least 80%) are people who are learning the concept for the first time (like it or not, that's the reality I'm afraid). Currently the only section that appeals to me in the article is the section on Graphs; the rest of the article is really basic material and I don't think that a professional mathematician will gain anything from it. There are people who want to read proofs and I see nothing wrong in having a different article on them. Merging the proofs here would go off the topic since the proofs are far too basic for research purposes. But I see nothing wrong in having another article on them (may I stress). Sorry to be so blunt but this is another instance where we are wasting time on WP and to be quite honest, I have not seen anything productive going on in the math project since the failed FA nomination of vector space. Oppose but Don't Delete the proofs. --PST 07:56, 23 February 2009 (UTC)

I changed the /Proofs page to a redirect to Locally connected space#Components and path components which outlines the basic ideas given there. The actual details are mostly trivial and including them on Wikipedia seems to be a violation of WP:NOTTEXTBOOK. The consensus here seems to be that the proofs shouldn't be included in this article and the /Proofs article does not have content which is notable on it's own, so changing to a redirect seems like the best option.--RDBury (talk) 18:26, 1 January 2010 (UTC)

Totally Disconnected Example

I don't see why the example given at the end of the Disconnected Spaces subsection is totally disconnected. In particular, how does one show that the two element set containing both the zeros is not a connected component? Does anyone have a reference for this example?

Thanks, John MacQ 82.28.178.242 (talk) 22:57, 20 December 2009 (UTC)

That two-point set is disconnected, as each zero is closed in the whole space and hence also in the two-point subspace. Algebraist 13:08, 21 December 2009 (UTC)

Perhaps I'm missing something, but this can't be enough on its own. If your argument shows that the set is disconnected, doesn't it also show (for instance) that any Hausdorff space is totally disconnected? John MacQ 82.28.178.242 (talk) 19:36, 21 December 2009 (UTC)

No. It shows that any two-point set in any Hausdorff space is disconnected. This is a necessary but not sufficient condition for a space to be totally disconnected. Algebraist 19:46, 21 December 2009 (UTC)

Ah okay I see the difference. So for a Hausdorff space the components have size either 1 or infinity. The reason I'm finding this so confusing is that people seem frequently to give the definition of Totally Disconnected as being the definition of Totally Separated given in this article. I have a reference that says the definitions agree on spaces that are in addition compact. But I can (I think) modify the example given here so that the space is compact and totally disconnected but not Hausdorff (take the subspace consisting of the zeros and the elements 1/n for each natural number n). John MacQ —Preceding unsigned comment added by 82.28.178.242 (talk) 20:27, 21 December 2009 (UTC)

First line in the article

"In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the union of two or more disjoint nonempty open subsets." Is this actually rigorous? I can express any set connected set as a union of disjointed subsets if I partition it right...Say the interval [0,1] is clearly connected. I can write it: [0,0.5) union [0.5,1] which are disjoint sets, contradicting the first line in the article. Or perhaps I am missing something - thoughts anyone? —Preceding unsigned comment added by 69.196.185.126 (talk) 00:54, 17 May 2010 (UTC)

Notice that it says "open". --Zundark (talk) 08:01, 17 May 2010 (UTC)

Captions don't currently match pictures

The captions at the top right don't match the pictures - for example the orange space D is connected, though the caption implies it is not. (It is however not simply connected). Rfs2 (talk) 13:05, 5 May 2011 (UTC)

I'm not sure what you mean. The caption says "The pink space C at top and the orange space D are both connected; however C is also simply connected whereas D is not" (emphasis added). This seems to agree with what you are saying, yes? (Of course, a separate question is whether the second picture really adds anything here or simply confuses the issue. Personally, I preferred the illustration before the second picture was added, and I think that the content in the caption about "simply connected" is not really germane to the topic of this article. We have a separate article on simply connected spaces, and I'm not sure that examples and discussion of simple connectedness are really helpful on the main picture in this article here.) —Caesura^(t) 14:59, 5 May 2011 (UTC)

I suspect your puzzledness, and especially mine, are due to this being the first time I've tried to do anything but reading in Wikipedia. Specifically, at the time I put my comment on the talk page, the captions to the two pictures both referred to both the coloured sets and their complements, made no reference to simply-connectedness, and were wrong. I had no desire to start referring to simply-connectedness, just to get somebody who knew what they were doing to fix the wrongness. Within five minutes (literally) of my posting my comment on the talk page, somebody had changed the captions to being correct (though now also referring to simply-connectedness). However, on viewing the history of the main page, I now cannot see any evidence of the version to which I commented. If somebody else had just put the wrong captions in before I posted my comment, and another person then rolled-back their edits, would the history of the two events then be removed ? (I suspect this discussion should be happening elsewhere - if you read this and know what you are doing, please remove this whole thread of discussion from here and answer my questions in the right place, which may perhaps be my user page ??) —Preceding unsigned comment added by Rfs2 (talk • contribs) 11:35, 6 May 2011 (UTC)

There was a version that is exactly as you described, but the wording was fixed on April 22—nearly two weeks before you made your comment. I'm not sure why you would have seen an April 21 version of the article on May 5, unless you were browsing around the article's history and got confused, or maybe there was some weird caching thing going on with your browser. (And to answer your question, standard rollback would show up in the page history. It is possible in principle for the revision history to be suppressed but this is extremely rare and probably not what happened here. The suppression tool is available only to a small number of Wikipedians and is used only for the purpose of expunging seriously problematic content, like phone numbers of anonymous contributors. Under normal circumstances, all editing activity, including rollback, is visible in the page history. And this talk page is a fine place for this thread; despite this derail, it's chiefly a discussion of the article.) Anyway, welcome to Wikipedia! —Caesura^(t) 12:39, 6 May 2011 (UTC)

I note the point above that this article may not be the place to discuss simple connectedness, but whether or not simple connectedness ought to be on the page, it is, but I'm not sure it is accurately stated. Barnsley's "Fractals Everywhere" p 28 gives an example of a metric space with a single "hole" in it similar to pink example 'C' in the sidebar. He says that this is not simply connected because paths that go around the 'hole' in one direction cannot be "continuously deformed" into paths that go in the other direction. In other words 'C' is multiply, not simply, connected. I am not trained in topology, so can't be certain I am reading him correctly, but it looks to me like this image contradicts Barnsley. Baon (talk) 18:26, 24 January 2013 (UTC)

The image is very misleading, but it shows a black line coming out of the right side of the hole. This line is meant to be a thin gap in the pink area (which is clearer if you click through to the larger version of the image), meaning that it's impossible to have a path going all the way around the hole. --Zundark (talk) 18:42, 24 January 2013 (UTC)

I see. I guess I thought that was just a pointer without a label... Thanks for clearing that up. Baon (talk) 20:04, 24 January 2013 (UTC)

I got tricked by the image also. I almost deleted it before this conversation revealed to me that the line wasn't meant to be of finite length. I suggest removal of the "C is also simply connected" stuff A.) because the image is ambiguous regarding the pink region's simply-connectedness B.) because it is unnecessary to mention simple-connectivity on the main image of the page and C.) because explaining the technicality that makes the pink region simply connected would add confusing details irrelevant to the fundamental subject of the page. 72.44.12.209 (talk) 21:38, 30 July 2013 (UTC)

Disjoint disks need not be open for union to be disconnected

Regarding "Other examples of disconnected spaces (that is, spaces which are not connected) include...the union of two disjoint open disks in two-dimensional Euclidean space." I think this is misleading since the disjoint disks need not be open (in $\mathbb {R} ^{2}$) in order for the their union to be open as a subspace (i.e. in the subspace topology).

Quinn (talk) 17:37, 27 December 2011 (UTC)

It's not true that the union of any two disjoint disks is connected, though; a closed disc and open disk that are tangent will have a connected union. So we can't just remove the word "open". We could try to expand the text to precisely characterize when the union would be disconnected, but that would take a lot of space and distract from the point of just giving an example. So it's easier to just leave in the hypothesis. The sentence does not say "if and only if", after all, so it's not wrong. — Carl (CBM · talk) 22:08, 27 December 2011 (UTC)

is the definiton correct?

It seems like words "relative topology" are missing from the definition. Without this specification the definition seems to be wrong. Consider an open interval $math$(0;1)$math$ and a point ${3}$. Single point in metric space is a closed set, thus you can never divide $math$(0;1) \Cup {3} $math$ into two open subsets in the original topology and should consider it connected. That is, obviously, a nonsence.

article formatting

I realize the guidelines of the Wikipedia math group don’t require it, but I have standardized the display of all mathematical symbols and expressions in the article to LaTeX. It was previously a mixture of the three typical fonts. The exception is the headers and captions of the images which I left unchanged, as some of the current fonts there were consistent with those within the image, which I cannot change — and the relative positioning of captions is tricky enough as it is. It is my feeling that consistency of font enhances readability and LaTeX is my personal preference — and what I know. But I am not especially invested in the matter, so anyone should feel free to revert any of it. Jmcclaskey54 (talk) 01:15, 9 June 2022 (UTC)

Open sets should redirect to https://en.wikipedia.org/wiki/Open_set#Notes_and_cautions

Definition 1 : X "is connected, that is, it cannot be divided into two disjoint non-empty open sets." This has a abuse of notation where instead of open topological spaces it says open sets . This has not been clarified so it should redirect to notes and caution in open sets. I don't know how to make this edit. This is why i am posting this in talk so someone else might. A topological space can be both open and closed.

This should also apply to first paragraph of the article 14.139.123.36 (talk) 06:19, 14 March 2023 (UTC)

"Locally path-connected space" listed at Redirects for discussion

The redirect Locally path-connected space 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 space until a consensus is reached. 1234qwer1234qwer4 19:20, 4 November 2023 (UTC)

Arc connectedness issues

@Kaba3: It seems that you added most of the contents for the section #Arc connectedness. It seems that a big part of it seems WP:OR, which as you know is not acceptable in Wikipedia. Do you have any specific references that can be used? In particular, referring to the condition that we care only about topologically distinguishable points. Also some of the statements mentioning an equivalence relation are mathematically incorrect. Looking forward to your comments. PatrickR2 (talk) 22:17, 3 February 2024 (UTC)

@PatrickR2 Well, there is no arc from a point to itself, so even if we restrict to T0 spaces, then the definition must at least say "there exists an arc between distinct points". The topological distinguishability generalizes distinctness to non-T0-spaces. Perhaps it is just my own extension. In that case it is possible to revert to assuming T0, and hence avoid WP:OR. Again, feel free to modify as you wish. Which statements are incorrect? Kaba3 (talk) 23:04, 3 February 2024 (UTC)

@Kaba3 Arc connectedness is usually defined as there exists an arc between any two points, where the case of two equal points is handled by the convention that a constant path is also an arc to make that work. That is used for example in Steen & Seebach. Which is the same as "there exists an arc between distinct points", like you said. Anyway, that's minor difference in presenting things. But no need to assume T0. From the definition, T0 spaces will not be arc connected. That's perfectly fine.

On the other hand, (even with the convention above about constant paths) the relation "x and y are joined by an arc" is reflexive and symmetric, but is not an equivalence relation. So much of what you wrote regarding that is not correct. Does the notion of arc-component even make sense? Please let me know what you think.

In any case, Wikipedia should only reflect well accepted sources and not have original material, even when that extra material is not incorrect. And all (most?) statements should ideally be backed by reliable sources. So if you have good sources for any of that, we should definitely add them. Otherwise, we can clean up some things. PatrickR2 (talk) 20:53, 5 February 2024 (UTC)

@PatrickR2 The relation corresponding to "there are arcs between indistinguishable points" would be "for points x and y, either x and y are indistinguishable, or there is an arc from x to y". But now that you mention it, I don't know whether it is always an equivalence relation. For $\Delta$-Hausdorff spaces, I believe it is by https://arxiv.org/pdf/2207.07242.pdf, because then path-connectedness and arc-connectedness coincide through loop-deletion. And in that case "arc-components" would be the path-components. Anyway, I agree that beyond those spaces the arc-connectedness section is currently on a shaky ground, and should be reverted to something citable. Kaba3 (talk) 21:26, 5 February 2024 (UTC)

Come to think of it, if the union of two arcs can be shown to be $\Delta$-Hausdorff, then that would prove the equivalence since the cited paper can be applied in the subspace of the union of the arcs. Kaba3 (talk) 21:55, 5 February 2024 (UTC)

For a simple example where the relation of joined by an arc is not transitive: Non-Hausdorff manifold#Line with two origins. It is T1, but not Delta-Hausdorff. And the union of two arcs in not Delta-Hausdorff in that space. PatrickR2 (talk) 00:24, 6 February 2024 (UTC)

@PatrickR2 Yes, when I had time to think it that's what I also came up with the exact same reasoning. Kaba3 (talk) 02:03, 6 February 2024 (UTC)

Thanks for the feedback. When I get some time, I'll edit a few things then and add references. PatrickR2 (talk) 03:20, 6 February 2024 (UTC)