This question is related to [question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?"](https://mathoverflow.net/questions/92206/) but is not exactly equivalent in my opinion. It is even suggested in one of the answers to [92206](https://mathoverflow.net/questions/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](https://mathoverflow.net/questions/92206/), but i wonder if there is more. Another related question: ["Topological Characterisation of the real line"](https://mathoverflow.net/questions/76134/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](http://terrytao.wordpress.com/2011/10/08/254a-notes-5-the-structure-of-locally-compact-groups-and-hilberts-fifth-problem/))?
* 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](https://math.stackexchange.com/questions/322411/), 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](https://mathoverflow.net/questions/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](https://mathoverflow.net/questions/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.



Topological characterization of the closed interval $[0,1]$
Consider the class of all Hausdorff compacts with distinct points (i.e. which have more than $1$ point) that are [absolute retracts](http://planetmath.org/absoluteretract) in the class
of Hausdorff compacts.
Then $[0,1]$ is up to homeomorphism the only member of this class that embeds into every other.


