Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:52:33Z http://mathoverflow.net/feeds/question/1454 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1454/is-there-a-co-hahn-mazurkiewicz-theorem-for-line-filling-spaces Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces? skupers 2009-10-20T17:05:17Z 2009-10-25T15:05:12Z <p>A famous theorem on space-filling curves is the Hahn-Mazurkiewicz theorem: Let X be a Hausdorff space, then there exists a surjective continuous map [0,1] \to X if and only if X is compact, connected, locally connected and metrizable. </p> <p>Is there a similar characterisation for all (Hausdorff) spaces having a surjective continuous map into the unit interval (which I decided to call line-filling spaces)?</p> http://mathoverflow.net/questions/1454/is-there-a-co-hahn-mazurkiewicz-theorem-for-line-filling-spaces/1969#1969 Answer by Konstantin Slutsky for Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces? Konstantin Slutsky 2009-10-22T22:37:54Z 2009-10-22T23:40:25Z <p>There is, probably, no hope for a similar characterization since for any topological space Y and any X, which is line-filling in your terminology, if f: X \to [0,1] is continuous and surjective then h: X \cap Y \to [0,1], where X \cap Y is a sum of spaces (=disjoint union) X and Y, given by h|_X = f and h(Y) = 0 is also surjective and continuous. So your spaces maybe as bizzare as you want.</p> <p>Update: Alejandro, thank you for your comment. I don't think though, that connectedness (or path-connectedness) would make things better. Again, for any connected (path-connected) Y and any line-filling space X, f: X \to [0,1] take x \in X and y \in Y and glue X and Y at the point (x,y) (that is consider equivalence relation with only one non-trivial equivalence x \equiv y and take the factor space). Denote the result by Z. Then one extends f to Z by defining f(Y) = f(x). I believe Z is then connected (if X and Y were) and again pretty random.</p> http://mathoverflow.net/questions/1454/is-there-a-co-hahn-mazurkiewicz-theorem-for-line-filling-spaces/1976#1976 Answer by Alejandro for Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces? Alejandro 2009-10-22T23:27:42Z 2009-10-25T15:05:12Z <p>Konstantin, you're right. But the disjoint union of spaces produces a non-connected space and well, I imagine, Skupers should be interested in characterizing connected spaces.</p> <p>UPDATE: After the second remark of Konstantin, I think we should reformulate the original question of Skupers asking about the characterization of connected "MINIMAL line-filling spaces", i.e. spaces which have no proper line-filling subspace. </p>