Question

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.

I am mostly interested in its significance for the study of Hausdorff compacts (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.

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Update. Related question: "Topological Characterisation of the real line".

Answer 1

In this edit, a PS is added at the end.

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).

Answer 2

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:

1. There exists a homeomorphism $f\colon [0,1]\to X$ such that $f(0)=a$ and $f(1)=b$.
2. Every connected subset of $X$ containing $\lbrace a,b\rbrace$ is equal to $X$.
3. The space $X$ is locally connected, and any connected and compact subset of $X$ which contains $\lbrace a,b\rbrace$ is equal to $X$.
4. For every $x\in X\setminus\lbrace a,b\rbrace$, the space $X\setminus\lbrace x\rbrace$ is not connected.

Answer 3

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).

Answer 4

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.