# Surniversal spaces

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?