**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,
*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?

chainablecontinuum, meaning one for which every open cover refines to a cover $U_1, \dots , U_n$ where $U_i \cap U_j \ne \emptyset$ iff $|i - j| \le 1$. This was proved by Mioduszewski in 1962. $\endgroup$ – John Baez Nov 20 '17 at 17:16