Topological characterization of the closed interval $[0,1]$ 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.
 A: A space which is homeomorphic to the closed unit interval is called a simple arc in the monograph "Dynamic topology" by Whyburn and Duda and there is a characterisation of it on p. 70 of this book.  This assumes that the given space  is a metric space, a condition which  can be avoided by using the Urysohn metrization theorem.  
On request, the characterisation is as follows:  a space is a simple arc if and only if  it is a non-degenerate (i.e., with more than one point) compact, connected set which is second countable and such that each point (with the exception of two specified ones---the endpoints) is a 
cut point.  (A point in a connected space is a cut point, if its complement is disconnected).
A: This is only a guess,   I don't know whether is true, but I find it plausible.  
Consider all the triplets $(X, x_0, x_1)$ where $X$  is a compact, second-countable, connected   Hausdorff space, and  $x_0,x_1$ are distinct  points in $X$.    We say that the triplet satisfies the property  $P$ if the following hold.


*

*For any $x\in X$, $x\neq x_0,x_1$  the complement $X\setminus \lbrace x\rbrace$ has exactly two connected components, each containing exactly   one of the points $x_0,x_1$. Denote by $C_i$ the component containing $x_i$, $i=0,1$.

*Set


$$\bar{C}_i:= C_i\cup\lbrace x\rbrace$$ 
Then each of the triplets $(\bar{C}_i, x_i,x)$ is homeomorphic to the triplet $(X, x_i, x_{1-i})$.
Here is my claim:  a triplet $(X.x_0,x_1)$  satisfies property $P$ if and only if  it is  homeomorphic to  the triplet $(\; [0,1],0,1\;)$
Acknowledgments.  I want to thank all the commenters for the useful explanations. (I've added second-countability to my claim which  turned out to be an old result.)   Here's an odd simple  observation which I find intriguing, and it may or may not be  useful.
Observe that on the space $\newcommand{\eT}{\mathscr{T}}$ $\eT$ of (homeomorphisms types of) second countable, compact, connected,  Hausdorff triplets $(X,x_0,x_1)$ there is a structure of associative semigroup
$$ (X, x_0,x_1)* (Y,y_0,y_1)= (Z,x_0,y_1), $$
where $Z$ is the space obtained by gluing  $X$ to $Y$ by identifying $x_1$ with $y_0$.   This operation is clearly involved in defining the fundamental group.
Denote by $\newcommand{\be}{\boldsymbol{e}}$ $\be$  the triplet $([0,1],0,1)$. Observe that $\be$ is an idempotent $\be\ast\be=\be$.  This fact alone makes possible the definition of the fundamental group.
The results      described  in the comments show  that $\be$ is  characterized  by the property
$$ \be =t_1\ast t_2,\;\;t_1,t_2\in \eT \Longleftrightarrow t_1=t_2=\be. $$ 
Here is an amusing question. Is it true that $\be$ is the unique idempotent of the semigroup $(\eT,\ast)$? I'm inclined to believe that  the answer is positive.
A: Consider the class of all Hausdorff compacts with distinct points (i.e. which have more than $1$ point) that are absolute retracts in the class
of Hausdorff compacts.
Then $[0,1]$ is up to homeomorphism the only member of this class that embeds into every other.
A: I'm not sure if this counts as a characterization, but it's a cool fact worth noting. If you start with $I$ and take the reflective subcategory it generates then you get the subcategory of completely regular spaces (aka Tychonoff spaces or $T_{3.5}$ spaces). These are very important for general topology and have good categorical properties, e.g. closure under subspaces, products, etc. The class of completely regular spaces coincides with the class of uniformizable spaces, and these have also been of historical importance.
On a related note, the coreflective subcategory generated by $I$ is the subcategory of $\Delta$-generated spaces. This forms a combinatorial model category, and has gained much interest in recent years as a place to do homotopy theory without having to worry about smallness issues. Now, as far as I know this doesn't characterize $I$, because maybe something else could generate the same subcategories. If someone knows whether or not such a generator has to be unique (or unique up to ---) I'd be interested. This is CW, so feel free to edit if you know this story and want to. I learned this from my advisor, Mark Hovey. It's somewhat far away from the work I do.
A: 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:

Up to isomorphism, the unit interval is the only complete totally ordered set (with end points) that has a continuous "betweenness function".

[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"].


Theorem. Up to homeomorphism, the unit interval is the only continuum $X$ (Hausdorff, connected, and compact) in which all but two points of $X$ are cuts points, and which additionally has the property that $X^2$ supports a continuous "betweenness" function (relative to the separation order).


A: The following topological characterization is close to that of the real line which is
indicated in another MO thread, but does not seem to have been pointed out here.
Let $X$ be a topological space, let $a,b$ be points of $X$ such that $a\neq b$. Assume that $X$ is compact, connected  and separable. The following conditions are equivalent:


*

*There exists a homeomorphism $f\colon [0,1]\to X$ such that $f(0)=a$ and $f(1)=b$.

*Every connected subset of $X$ containing $\lbrace a,b\rbrace$ is equal to $X$.

*The space $X$ is locally connected, and any connected and compact subset of $X$ which contains $\lbrace a,b\rbrace$ is equal to $X$.

*For every $x\in X\setminus\lbrace a,b\rbrace$, the space $X\setminus\lbrace x\rbrace$ is not connected.

