Talk:Contractible space

Hi there,

Why isn't there a link on the left to the french version? The french version already exists: here is a link to it:

http://fr.wikipedia.org/wiki/Espace_contractile

Could someone add the link, I don't know how to do it. —Preceding unsigned comment added by 78.120.162.191 (talk) 11:16, 11 April 2010 (UTC)[reply]

Done

It's ok, I've added the link

The empty space contractible?

Is the empty space contractible? My impression is that when one claims that some (sub-)space is contractible, then this includes that it it nonempty. (So this is not just an academic question.)

The empty space X fulfils the formal definition given here, since identity map on X is homotopic to some constant map. (There is only one map on X, and it is a constant map). But of course it is not homotopy equivalent to a one-point space. (one of the "equivalent" definitions given in the article). On the other hand, any two maps f,g: Y → X are homotopic. (Such a map exists iff Y is also empty.) So the different "equivalent characterizations" are contradictory on this question.

If "The cone on any space X is always contractible" then the empty space (which is the cone on the empty space) should be contractible, on the other hand. (Maybe this example should be added to the article cone (topology).

What do the textbooks say?

some findings on the internet about this question

http://arxiv.org/abs/1004.4168 in the abstract: "We prove that this complex is contractible, which was conjectured by .... More generally, the ... is contractible or empty."

These authors believe that the empty set is not contractible.

The first page of http://www.emis.de/journals/HOA/IJMMS/Volume24_8/185093.pdf (Lemma 2.1) says "let F(J) be a nonempty contractible subset of X..."

These authors believe that a contractible set might be empty.

http://www.sciencedirect.com/science/article/pii/S0166864100000742, Intersection properties of Helly families, Władysław Kulpa, Topology and its Applications, Volume 116, Issue 2, 15 November 2001, Pages 227–233 write in the abstract: "... is a nonempty contractible set,"

This author believes that a contractible set might be empty.

http://ncatlab.org/nlab/show/contractible+space says: "Sometimes one allows also the empty object ∅ to be contractible. To distinguish this, we say ..."
http://mathforum.org/kb/message.jspa?messageID=5657021 says it is a matter of taste.

So maybe there are divided opinions or different usages, and the article should not decide it but add a word of caution about the potential different usages.

GuenterRote (talk) 18:58, 19 January 2013 (UTC)[reply]

Closed curves

The article lead states that, "Intuitively, a contractible space is one that can be continuously shrunk to a point." The Jordan curve theorem allows that any closed loop on a locally Euclidean surface can be continuously shrunk to a point. Examples E and F in the figure and caption shown in the article lead, and reproduced here, suggests that this is untrue. I guess that in the two usages of the phrase, being "continuously shrunk to a point" means something subtly different. Can this be clarified in the article, please? — Cheers, Steelpillow (Talk) 07:38, 19 January 2016 (UTC)[reply]

Oh, that is a problem! I'll fix that. ProboscideaRubber15 (talk) 20:10, 2 April 2016 (UTC)[reply]