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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).
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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
257:, 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,"
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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
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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.)
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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.
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in the abstract: "We prove that this complex is contractible, which was conjectured by .... More generally, the ... is contractible or empty."
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Illustration of some contractible and non-contractible spaces. Spaces A, B, and C are contractible; spaces D, E, and F are not.
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The article lead states that, "Intuitively, a contractible space is one that can be continuously shrunk to a point." The
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Why isn't there a link on the left to the french version? The french version already exists: here is a link to it:
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says: "Sometimes one allows also the empty object ∅ to be contractible. To distinguish this, we say ..."
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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point" means something subtly different. Can this be clarified in the article, please? — Cheers,
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should not decide it but add a word of caution about the potential different usages
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So maybe there are divided opinions or different usages, and the article
244:(Lemma 2.1) says "let F(J) be a nonempty contractible subset of X..."
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http://www.sciencedirect.com/science/article/pii/S0166864100000742
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These authors believe that a contractible set might be empty.
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These authors believe that the empty set is not contractible.
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This author believes that a contractible set might be empty.
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http://www.emis.de/journals/HOA/IJMMS/Volume24_8/185093.pdf
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Could someone add the link, I don't know how to do it.
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http://mathforum.org/kb/message.jspa?messageID=5657021
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