I would like to learn purely topological characterizations of the closed real intervals (to justify the existence of algebraic topology). In particular, such a characterization should not use real numbers. I would like to see in what sense $[0, 1]$ is topologically more important than, for example, the set of rationals, or the complex unit disc. I am especially interested in why the unit interval is so important in the study of compact Hausdorff spaces (metrization, Urysohn's lemma).
I have come up with the following one.
Start with a definition of path connectedness that does not use $\mathbb R$: $x_1$ and $x_2$ are connected by a path in $X$ if for every Hausdorff compact $C$ and $a,b\in C$, there is a continuous $f\colon C\to X$ such that $f(a)=x_1$ and $f(b)=x_2$. Now, if $X$ is a Hausdorff space and distinct points $x_1$ and $x_2$ are connected by a path in $X$, then there is a minimal subspace of $X$ in which $x_1$ and $x_2$ are still connected by a path, and every such subspace is homeomorphic to $[0, 1]$.
Roughly speaking, $([0, 1], 0, 1)$ is the minimal bi-pointed Hausdorff space such that every bi-pointed Hausdorff compact can be mapped into it with the distinguished points being sent onto the distinguished points.
Maybe in some sense it can be also said that $X = [0, 1]$ is the "minimal" Hausdorff space such that every Hausdorff compact embeds into $X^N$ for some $N$, but i do not know how to make this precise.
My question is: what are other "natural" topological characterizations of $[0, 1]$?
Update: I have duplicated this question on MathOverflow.