It happens to everyone. First, you learn real analysis and suddenly you understand what it means to do rigorous mathematics. You learn the precise definition of what a continuous function is; you learn that if they are defined on a closed bounded interval then they are bounded and attain their bounds; you learn about things called ‘Cauchy sequences’ and ‘The Bolzano-Weierstrass Theorem’.
Maybe you even start to think about doing analysis in \(\mathbb R^n\), but you then quickly learn that this is but one example of something called a ‘metric space’. New avenues open. You generalize the definition of continuity. You learn about the ‘contraction mapping theorem’ and how to apply that to the study of differential equations.
Then the whole thing gets turned on its head when you learn what a topological space is. Suddenly, you have a vastly more general definition of what it means for something to be continuous. You realize that all the things you learned in real analysis were really worked examples of topological facts. Continuous functions on a closed bounded interval are bounded and attain their bounds because the continuous image of a compact space is compact. \(\mathbb R^n\)? You know what \(\mathbb R\) is and you know what the product topology is, so what more do you need to know?
Then you start to learn algebraic topology and suddenly closed bounded intervals of real numbers are all over the place again. \([0,1]\) is a crucial part of the definition of a path or a homotopy. Maybe you start proving theorems that apply not to all topological spaces, but to CW complexes: i.e., spaces built up in a fairly complicate way from unit balls in \(\mathbb R^n\). It’s not unreasonable to feel a little cheated. Wasn’t it the whole point of topology that we didn’t have to deal with \([0,1]\) any more? Why is it everywhere suddenly?
Now, it’s true that every topological space admits a weak homotopy equivalence with some CW complex. But that doesn’t answer our question. Firstly, the definition of a weak homotopy equivalence involves \([0,1]\) anyway. And, anyway, why shouldn’t we be able to come up with some alternative definition of ‘CW complex’, based on some different fundamental topological space, and prove a similar result for that. What’s so special about \([0,1]\)?
Pathing systems
Let’s come back to the first definition we tend to meet in topological that relies on \([0,1]\) in an essential way. When we learn topology, we learn two non-equivalent definitions of what it means for a space to be connected. One definition is purely topological: a space \(X\) is connected if it cannot be written as the union of two disjoint non-empty open sets.
For the second definition, we define a path from a point \(x\) to a point \(y\) in a space \(X\) to be a continuous function \(\gamma \colon [0,1] \to X\) such that \(\gamma(0) = x\) and \(\gamma(1) = y\). We then say that \(X\) is path connected if any two points in \(X\) can be joined by a path in this way.
That raises the question of what would happen if we replaced \([0,1]\) with some other space in the above definition. Would we still get a sensible definition of connectedness?
There are some silly choices of space to use: if we use \({0, 1}\), then every space is connected. If we use \({0}\), then only empty or singleton spaces are.
But let’s not forget that \([0,1]\) has some useful properties that make it a good choice for defining paths. In particular, if we have paths \(\gamma,\gamma’ \colon [0,1] \to X\) such that \(\gamma\) is a path from \(x\) to \(y\) and \(\gamma’\) a path from \(y\) to \(z\), then we can come up with a path \(\gamma;\gamma’ \colon [0,1] \to X\) that is a path from \(x\) to \(z\). We do this by defining \(\gamma;\gamma'(t) = \gamma(2t)\) if \(t<0.5\) and \(\gamma;\gamma'(t) = \gamma'(2t-1)\) if \(t\ge 0.5\).
It makes sense, then, to insist that our new candidate spaces have similar properties. That is, we define a pathing system to be given by a space \(J\), together with distinguished points \(j_0,j_1\in J\), and, for any space \(X\) and any pair of continuous functions \(\gamma,\gamma’ \colon J \to X\) such that \(\gamma(j_1) = \gamma'(j_0)\), a continuous function \(\gamma;\gamma’ \colon J \to X\) such that \(\gamma;\gamma'(j_0) = \gamma(j_0)\) and \(\gamma;\gamma'(j_1) = \gamma'(j_1)\).
I shall also impose one more condition to make sure that the choice of \(\gamma;\gamma’\) is at least fairly well-behaved. If we have a continuous function \(\phi \colon X \to Y\), where \(Y\) is some other space, then we should insist that
$$
\phi \circ (\gamma;\gamma’) = (\phi \circ \gamma);(\phi \circ \gamma’)
$$
as continuous functions \(J \to Y\).
From the perspective of category theory, this kind of requirement is known as naturality: it says that a certain collection of morphisms forms a natural transformation. You might like to work out the details.
Multiple concatenations
Of course, we are not limited to composing together two paths. For example, if we have four paths, then we can compose them in pairs to get two paths, and then compose those paths together to get a single path. In a similar way, we can define the \(2^n\)-fold composition of \(2^n\) paths inductively as follows. The bifold composition is the usual composition. Then the \(2^{n+1}\)-fold composition of \(2^{n+1}\) paths (with matching endpoints) is formed by splitting into two equal collections of \(2^n\) paths, forming the \(2^n\)-fold composition of each and then composing the two paths thus formed. Equivalently, we could split our collection of \(2^{n+1}\) paths into \(2^n\) pairs of paths, compose each pair to form \(2^n\) single paths, and then take the \(2^n\)-fold composition.
Now, given a pathing system \((J, j_0, j_1)\), write \(\bigwedge_{2^n} J\) for the space formed by taking \(2^n\) disjoint copies of \(J\) and then passing to the quotient space given by identifying \(j_1\) in copy \(i\) with \(j_0\) in copy \(i+1\). There are \(2^n\) obvious continuous functions \(J \to \bigwedge_{2^n} J\) whose endpoints match up, and we can take the \(2^n\)-fold composition of these paths to get a single continuous function
$$
p_n \colon J \to \bigwedge_{2^n} J
$$
such that \(p_n(j_0)\) is \(j_0\) in the first copy of \(J\) and \(p_n(j_1)\) is the \(j_1\) in the last copy of \(J\).
Now let us number the copies of \(J\) in binary from \(\underbrace{0\cdots0}_n\) to \(\underbrace{1\cdots1}_n\). I claim that if \(x\in J\) and \(m<n\), then the sequence of binary digits corresponding to the copy of \(J\) that \(p_m(x)\) is contained in is a prefix of the sequence of binary digits corresponding to the copy of \(J\) that \(p_n(x)\) is contained in. Indeed, by the definition of \(2^k\)-fold composition, \(p_n\) may be regarded as the \(2^m\)-fold concatenation of \(2^{n-m}\) paths \(J \to \bigwedge_{2^m}J\).
This means that, as \(n\) gets larger and larger, we get an infinite binary sequence corresponding to each \(x\in J\). We define a function \(\phi \colon J \to [0,1]\) by sending each \(x\in J\) to the real number whose binary expansion is given by that sequence.
\(\phi\) is easily seen to be continuous: indeed, the preimage of any set of the form \([0, q/2^n)\) is precisely the preimage under \(p_n\) of some open set in \(\bigwedge_{2^n}J\), and is therefore open in \(J\), and similarly for any set of the form \((q/2^n, 1]\). Moreover, we certainly have \(\phi(j_0) = 0\) and \(\phi(j_1) = 1\).
This means that if there is a \([0,1]\)-path from \(x\) to \(y\) in a space \(X\), then there is a \(J\)-path from \(x\) to \(y\) for any pathing system \((J, j_0, j_1)\). In other words, \([0,1]\) is special, because it gives us the most restrictive possible notion of path-connectedness while still allowing basic notions like composition of paths.
Universality of \([0,1]\)
Are there any other spaces with this special property? It’s fairly easy to show that there aren’t. First, notice that the function \(\phi\) that we constructed above makes the following diagram commute.
$$
\require{AMScd}
\begin{CD}
J @>{\phi}>> [0,1]\\
@V{p_2}VV @VV{_\times 2}V \\
J \wedge J @>{\phi\wedge\phi}>> [0,1] \wedge [0,1]
\end{CD}
$$
Moreover, \(\phi\) is in fact the unique continuous function \(J \to [0,1]\) with this property. In category theory, we say that \([0,1]\) is the final coalgebra for the functor \(\_ \wedge \_\). A general fact about final coalgebras is that they are unique up to isomorphism. You might like to deduce for yourself that any space \(I\) having the same property we’ve found for \([0,1]\) is in fact homeomorphic to \([0,1]\).
Other directions
There are a few other generalizations we could make to our notion of a ‘path’. We should be careful to avoid being too general: for example, we could define a path from \(a\) to \(b\) in a space \(X\) to be a connected subspace of \(X\) that contains \(a\) and \(b\) — then we would end up with the usual definition of connectedness.
One thing we might try is not to insist that every path is a continuous function out of the same space \(J\): so that the composition of two paths could have a different domain from either of its constituent parts. I’ll save that discussion for another time.