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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.

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)?

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2 Answers 2

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.

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.

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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.

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.

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There are no "MINIMAL line-filling spaces". If $f:X \to [0,1]$ is onto then $f^{-1}([0,1/2])$ is a proper line-filling subspace of $X$. –  Ramiro de la Vega Feb 13 '13 at 17:17
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