Theorem 2 (Shelah[^1])

Let X be a connected Polish space which is locally connected by arcs, and let $x$ be a point of X. If X is not unravelable, the group $\pi_1(X, x)$ has the power of the continuum.

Let $d$ be a metric defining the topology of X for which it is complete. Suppose that X is not unravelable. Then there exists a point $a\in X$ and, for every integer $n\geqslant 0$, a loop $c_n$ at $a$ in X whose image has diameter $\leqslant 2^{-n}$ and whose class in $\pi_1(X, a)$ is nontrivial.

Let us denote by K the set $\{0,1\}^{\mathbf{N}}$ and endow it with the product topology, the space $\{0,1\}$ being endowed with the discrete topology. For every element $\varepsilon = (\varepsilon_n)$ of K, let $c_{\varepsilon}$ be the mapping of $\mathbf{I}$ into X defined by $c_{\varepsilon}(0) =a$ and, for $2^{-n-1}\leqslant t\leqslant 2^{-n},c_{\varepsilon}(t) =c_n(2^{n+1}t-1)$ if $\varepsilon_n= 1$ and $c_{\varepsilon}(t) =a$ otherwise. One has $c_{\varepsilon}(0) =c_{\varepsilon}(1) =a$. The mapping $c_{\varepsilon}$ is continuous at every point $t\in ]0,1]$. It is also so at 0, since $d(a, c_{\varepsilon}(t))\leqslant 2^{-n}$ if $t\in [0,2^{-n}]$. The mapping $c_{\varepsilon}$ is therefore a loop in X at $a$.

If $\varepsilon$ and $\varepsilon '$ are elements of K such that $(\varepsilon_0, . . . , \varepsilon_n) = (\varepsilon '_0, . . . , \varepsilon '_n)$, then $c_{\varepsilon}(t) =c_{\varepsilon'}(t)$ for every $t\in [2^{-n-1},1]$ and $d(c_{\varepsilon}(t), c_{\varepsilon'}(t))\leqslant$ $d(a, c_{\varepsilon}(t)) +d(a, c_{\varepsilon'}(t))\leqslant 2^{-n}$ if $t\in [0,2^{-n-1}]$. It follows that the mapping $\varepsilon \mapsto c_{\varepsilon}$ of K into the space $\Omega_a(X)$ is continuous, when the space $\Omega_a(X)$ is endowed with the topology of compact convergence.

Let $\Gamma \subset K\times K$ denote the set of pairs $(\varepsilon , \varepsilon ')$ such that $c_{\varepsilon}$ is strictly homotopic to $c_{\varepsilon'}$. It is the graph of an equivalence relation R in K.