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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?
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This problem has been answered by Waraszkiewicz in 1934 who proved that no metric continuum is suruniversal for all planar continua. This result was later developed by Bellamy, Krasinkiewicz, Minc, Ingram, see Section 20 in the paper [T.Mackowiak, E.D.Tymchatyn, Continuous mappings on continua, II, Dissert. Math. 225 (1984) 57p].

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  • $\begingroup$ Thank you. (I've even written about a variation of Waraszkiewicz's spirals on MathOverflow). $\endgroup$ – Włodzimierz Holsztyński May 1 '16 at 8:26

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