Lemma 5

The set Γ is a meagre subset of $K\times K$.

Let Z denote the space $K\times K\times \mathscr{C}_c(\mathbf{I}\times \mathbf{I}; X)$. The topology of the space $\mathscr{C}_c(\mathbf{I}\times \mathbf{I}; X)$ is defined by the distance $\delta$ given by $\delta (h, h') =$ sup$_{u\in\mathbf{I}\times\mathbf{I}}d(h(u), h'(u))$ and it is complete for this distance (TG, X, p. 20, corollaire and TG, X, p. 9, cor. 1); after TG, X, p. 24, th. 1, this topology is of countable type. The space $\mathscr{C}_c(\mathbf{I}\times \mathbf{I}; X)$ is therefore a Polish space. The same is true of the space Z, for K is a Polish space (TG, IX, p. 57, prop. 1).

Let H be the subset of Z consisting of the triples $(\varepsilon , \varepsilon ', h)$ such that $h$ is a strict homotopy joining $c_{\varepsilon}$ to $c_{\varepsilon'}$. The mappings from Z into $X^2$ given by $a_t: (\varepsilon , \varepsilon ', h)\mapsto (c_{\varepsilon}(t), h(t,0)),b_t: (\varepsilon , \varepsilon ', h)\mapsto$ $(c_{\varepsilon'}(t), h(t,1))$ and $c_t: (\varepsilon , \varepsilon ', h)\mapsto (h(0, t), h(1, t))$ are continuous, for every $t\in \mathbf{I}$, because the mapping $\varepsilon \mapsto c_{\varepsilon}$ from K into $\Omega_a(X)$ is continuous, and analogously the mappings $h\mapsto h(s, t)$ from $\mathscr{C}_c(\mathbf{I}\times \mathbf{I}; X)$ into X.

By definition, H is the intersection of the sets $\overset{-1}{a_{t}}(\Delta_X),\overset{-1}{b_{t}}(\Delta_X)$ and $\overset{-1}{c_{t}}((a, a))$, for $t\in \mathbf{I}$, where $\Delta_X$ denotes the diagonal of X. This proves that H is a closed subset of Z.

It follows that H is a Polish space. Let $p: Z\rightarrow K\times K$ be the canonical projection; we have $\Gamma =p(H)$ by definition. Since $K\times K$ is separated, Γ is a Souslin subset of $K\times K$ (TG, IX, p. 59, déf. 2). After IV, p. 362, prop. 7, it is an approachable subset of $K\times K$.

Suppose that Γ is not meagre. The space $K\times K$ is a compact topological space and therefore a Baire space (TG, IX, p. 55, th. 1). Endow the group $G_0$ of permutations of the set $\{0,1\}$ with the discrete topology; let the product topological group $G = G^{\mathbf{N}}_0$ operate diagonally in $K =\{0,1\}^{\mathbf{N}}$. The group G then operates continuously in $K\times K$ by the mapping $(g,(x, y))\mapsto (x, g\cdot y)$. After Lemma 4, the set V of elements $g\in G$ such that $\Gamma \cap g\cdot \Gamma =\not\emptyset$ is a neighbourhood of the identity element of G.

Let $g\in V$; let $(\varepsilon , \varepsilon ')\in \Gamma \cap g\Gamma$. Since $(\varepsilon , \varepsilon ')\in \Gamma$, we have $R\{\varepsilon , \varepsilon '\}$; since $(\varepsilon , \varepsilon ')\in g\Gamma ,g^{-1}\cdot (\varepsilon , \varepsilon ') = (\varepsilon , g^{-1}\varepsilon ')\in \Gamma$, so that $R\{\varepsilon , g^{-1}\varepsilon '\}$. We have thus proved that, for every $g\in V$, there exists $\varepsilon \in K$ such that $\varepsilon$ and $g\varepsilon$ are equivalent for R.

For $m\in \mathbf{N}$, let $\tau_m$ denote the element of G all of whose terms are equal to $e$ except that with index $m$, which is equal to the nontrivial element $\tau$ of $G_0$. There exists an integer $m$ such that $\tau_m$ belongs to V; let then $\varepsilon \in K$ be such that $\varepsilon$ and $\varepsilon '=\tau_m\cdot \varepsilon$ are equivalent for R. This implies that the loops $c_{\varepsilon}$ and $c_{\varepsilon'}$ are strictly homotopic. By construction, these loops coincide on the intervals $[0,2^{-m-1}]$ and $[2^{-m},1]$; on the interval $[2^{-m-1},2^{-m}]$, one is the constant mapping with image $a$ and the other is the mapping $t\mapsto c_m(2^{m+1}t-1)$. It follows that $c_m$ is strictly homotopic to the constant loop with image $\{a\}$, hence a contradiction. Lemma 5 is thus proved.

Let us now complete the proof of theorem 2. By Prop. 8, there exists a continuous injective mapping $\gamma$ of $\{0,1\}^{\mathbf{N}}$ into K whose image meets each equivalence class for R in at most one point. If $k$ and $k'$ are distinct elements of $\{0,1\}^{\mathbf{N}}$, the loops $c_{\gamma(k)}$ and $c_{\gamma(k')}$ are not strictly homotopic in X, and the mapping $\{0,1\}^{\mathbf{N}}\rightarrow \pi_1(X, a)$ given by $k\mapsto [c_{\gamma(k)}]$ is injective. In particular, Card($\pi_1(X, a)$)$\geqslant$ Card($\{0,1\}^{\mathbf{N}}$) $=$ Card($\mathfrak{P}(\mathbf{N})$). Since X is a metrizable topological space of countable type, the same is true of $\Omega_a(X)$ (TG, X, p. 24, th. 1). Hence $\Omega_a(X)$ is homeomorphic to a subspace of $[0,1]^{\mathbf{N}}$ (TG, IX, p. 18, prop. 12) and

Card(Ω$_a(X)$)$\leqslant$ Card([0$,1]^{\mathbf{N}}$) $=$ Card($\mathfrak{P}(\mathbf{N})^{\mathbf{N}}$)

= Card($\mathfrak{P}(\mathbf{N}\times \mathbf{N})$) $=$ Card($\mathfrak{P}(\mathbf{N})$).

A fortiori, Card($\pi_1(X, a)$)$\leqslant$ Card($\mathfrak{P}(\mathbf{N})$). It then follows from Cor. 2 of E, III, p. 25, that $\pi_1(X, a)$ has the power of the continuum, which was to be proved.