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)?
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 $h: X \sqcup Y \to [0,1]$, where $X \sqcup Y$ 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 bizarre 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.
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.
I stumbled across this thread and looked into it a bit more. This is the most relevant paper that I found and covers some of the reasonable counterexamples when this problem is approached from the viewpoint of regular spaces (since completely regular spaces admit such maps, by definition).
Krzysztof Chris Ciesielski and Jerzy Wojciechowski, Cardinality of regular spaces admitting only constant continuous functions, Top. Proc. 47 (2016) pp. 313-329, http://math.wvu.edu/~kcies/prepF/122.ConnectedRegularTopProcAccepted.pdf
Spaces like the Cantor set, the long line and the pseudo-arc are some starting points for thinking about this problem further. The relevant area for this question is Continuum Theory and in particular this problem would most naturally fall under the study of Hausdorff Continua.
A classification of Hausdorff spaces that have a completely regular quotient would probably be a relevant question to ask.
