# Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces?

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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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