This question is related to question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?" but is not exactly equivalent in my opinion. It is even suggested in one of the answers to 92206 that "there is nothing fundamental about the unit interval," but i would like to know what is fundamental about the unit interval. I have learned some answers from the answers to 92206, but i wonder if there is more. Another related question: "Topological Characterisation of the real line".
The question is:
What would be a "natural" topological characterization of the closed interval $[0,1]$?
Motivating questions (please do not answer them):
- Why is all the algebraic topology built around it?
- Why does it appear (in the guise of Lie groups) in the study of general locally compact topological groups (Gleason-Yamabe theorem)?
- Is it really necessary to define it algebraically as a subset of a field ($\mathbb R$) just to use it as a topological space?
I am mostly interested in the significance of $[0, 1]$ for the study of Hausdorff compacts (like metrizability, Urysohn's lemma).
I have some ideas and have asked already on Math.StackExchange, but decided to duplicate here.
One purely topological way to define $[0, 1]$ up to homeomorphism would be to define path connectedness first: $x_1$ and $x_2$ are connected by a path in a topological space $X$ 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 $f(b)=x_2$. Then it can be said that in every Hausdorff space $X$ with $x_1$ and $x_2$ connected by a path, every minimal subspace in which $x_1$ and $x_2$ are still connected by a path is homeomorphic to $[0, 1]$.
Maybe in some sense it can be said that $[0, 1]$ is the "minimal" Hausdorff space $X$ such that every Hausdorff compact embeds into $X^N$ for some $N$, but i do not know if this can be made precise.
In one of the answers to 92206 it was stated that $([0, 1], 0, 1)$ is a terminal object in the category of bipointed spaces equipped with the operation of "concatenation." This is the kind of answers i am interested in. As 92206 was concerned mostly with the fundamental group and tagged only with [at.algebraic-topology] and [homotopy-theory], i am asking this general topological question separately.
Best Answer
Not too long ago (2005) Harvey Friedman announced an attractive, novel characterization of the unit interval that seems to be little known, and might be the kind of answer you are looking for:
[Since arithmetical operations are continuous, it is clear that there are lots of continuous betweenness functions on the unit interval].
Here is the official characterization:
Theorem (H. Friedman). Let $X$ be a linearly ordered set with left and right endpoints and the least upper bound property. The following two statements are equivalent:
(a) $X$ is isomorphic to the usual closed unit interval.
(b) There is some $f:X^{2}\rightarrow X$ that is continuous relative to the order-topology on $X$, and $x < f(x,y) < y$ whenever $x < y$.
Friedman's proof appears in this FOM posting.
PS: Friedman's proof, when coupled with usual techniques of imposing an order on a continuum with at most two non-cut points (the "separation order") yields the following purely topological characterization [the main new idea being: in the classical characterization of the unit interval as the unique second countable continuum with exactly two non-cut points, "second countable" can be swapped with "supports a continuous betweenness function relative to the separation order"].