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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.
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
"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 to denote a topological space in a formal definition instead of merely. I would say the following definition is better than the current one:
is a connected topological space if and only if has only two subsets that are both open and closed (clopen) which are the and the entire X.
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
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)
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)
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 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)
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)
I'd like to have the ability to have a cross reference button to search for antonyms or opposites of words.
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)
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)
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)
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)
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)
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)
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)
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)
"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)
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 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)
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 ) 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 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.
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)
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)
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. 1234qwer 1234qwer 4 19:20, 4 November 2023 (UTC)
@ 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)
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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.
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
"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 to denote a topological space in a formal definition instead of merely. I would say the following definition is better than the current one:
is a connected topological space if and only if has only two subsets that are both open and closed (clopen) which are the and the entire X.
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
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)
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)
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 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)
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)
I'd like to have the ability to have a cross reference button to search for antonyms or opposites of words.
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)
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)
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)
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)
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)
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)
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)
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)
"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)
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 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)
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 ) 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 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.
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)
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)
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. 1234qwer 1234qwer 4 19:20, 4 November 2023 (UTC)
@ 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)