Talk:Connected space

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Mathematics rating: B Class Mid Priority  Field: Topology

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
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

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)

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)

Contents

[edit] 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)

[edit] 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)

[edit] "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. --LambiamTalk 07:56, 16 May 2006 (UTC)

[edit] 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"? --LambiamTalk 11:04, 24 July 2006 (UTC)

[edit] 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)

[edit] 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)