Proposition 9

Let X be an unravelable topological space and let $\mathscr{W}$ be a basis for the topology of X. For every point $x$ of X, the cardinal of the group $\pi_1(X, x)$ is bounded above by sup(Card($\mathscr{W}$), Card($\mathbf{N}$)). In particular, the Poincaré group of a metric space of countable type and unravelable is countable.

In fact, the space $\lambda_x(X)$ endowed with the endpoint mapping is a connected covering of X whose fibre at $x$ is $\pi_1(X, x)$. The assertion then results from I, p. 40, th. 3.