Basic background
On one hand there is a complete result: $\,\ $for every non-negative integer $n$ there exists an $n$-dimensional compact metric space $M^n$ such that it contains a homeomorphic image of every $n$-dimensional metric separable space (hence of every compact too).
On the other hand there is a partial result: $\,\ $for every non-empty compact metric space $\ X\ $ there exists a continuous surjection $\ f : D^{\aleph_0}\rightarrow X$.
Here, $\ D\ $ stands for the 2-point discrete space.
Of course, the two results stated above are very classical.
Definition
Let $\ S(X\ Y)\ $ be the set of all continuous surjections of a topological space $\ X\ $ onto a topological space $\ Y.\ $ A topological space $\ X\ $ is called surniversal for a class of topological spaces $\ \Omega\ $ if $\ S(X\ Y)\ne \emptyset\ $ for arbitrary $\ Y\in\Omega.$
Thus $\ D^{\aleph_0}\ $ is surniversal for the class of all compact metric spaces.
Main question
Does there exist a connected $1$-dimensional compact metric space which is surniversal for all connected compact metric spaces?
Diluted questions
- Does there exist a connected $1$-dimensional compact metric space which is surniversal for all $1$-dimensional connected compact metric spaces?
- Does there exist a connected compact metric space which is surniversal for all connected compact metric spaces?
