**Fact 1.** The Cantor set $K$ is "universal" among nonempty compact metric spaces in the following sense: given any nonempty compact metric space $X$, there exists a continuous surjection $f\colon K \to X$.  

**Fact 2.** The closed interval $I$ has a similar "universal" property among nonempty compact connected and locally connected metric spaces: given any such space $X$, there exists a continuous surjection $f \colon I \to X$.

This makes me wonder: **is there a compact connected metric space $J$ such that for any nonempty compact connected metric space $X$, there exists a continuous surjection $f \colon J \to X$?**

Such a space $J$, if it exists, would be 'intermediate' between $I$ and $K$: there would need to be continuous surjections 

$$    K \to J \to I $$

Fact 1 is sometimes called the **Alexandroff–Hausdorff theorem**, since appeared in the second edition of Felix Hausdorff’s _Mengenlehre_ in 1927 and also in an article by Pavel Alexandroff published in _Mathematischen Annalen_ in the same year.  Fact 2 was proved by Hans Hahn in 1914 and reproved by him more nicely in 1928.   For a nice history of these results, see:

* L. Koudela, [The Hausdorff–Alexandroff theorem and its application in theory of curves](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.186.8605&rep=rep1&type=pdf), _WDS'07 Proceedings of Contributed Papers_, Part I, 2007, pp. 257–260.

One may rightly complain that "universal" is the wrong word above, since we're _not_ claiming there exists a _unique_ continuous surjection, and indeed there's usually not.  A better term is **versal**.  There can be two non-homeomorphic spaces having the same versal property.  For example, $I^2$ would work just as well as $I$ in Fact 2, thanks to the existence of space-filling curves.

Nonetheless we can create a category in which these versal properties become universal, by a cheap trick.  Let $\mathrm{CompMet}$ be the collection of all homeomorphism classes of nonempty compact metric spaces, and put a partial order on this where $[X] \ge [Y]$ iff there exists a continuous surjection $f \colon X \to Y$.   The homeomorphism class of the Cantor set is the top element of the poset $\mathrm{CompMet}$.  My question asks if the subset of $\mathrm{CompMet}$ coming from connected compact metric spaces has a top element.  

I'd also appreciate any interesting information on this poset $\mathrm{CompMet}$.  

For example, I think that there's a map sending each element of $\mathrm{CompNet}$ to its number of connected components, and I think that this is an order-preserving map from $\mathrm{CompMet}$ to the cardinals less than or equal to the continuum.  But there also seems to be an order-preserving map sending each element of $\mathrm{CompNet}$ to its number of path-connected components.  Are there other interesting maps like this?