Theorem 1

Let B be a connected topological space, locally connected by arcs, and let $b$ be a point of B. The following properties are equivalent:

(i) The space B is unlaceable.

(ii) There exists a nonempty covering of B which is simply connected by arcs.

(iii) The space $\lambda_b(B)$, endowed with the endpoint mapping $\varepsilon_B:\lambda_b(B)\rightarrow B$, is a covering of B.

(iv) The group $\pi_1(B, b)$ is discrete for the admissible topology.

(v) The group $\pi_1(B, b)$ is discrete for the quotient topology of the compact-convergence topology.

Moreover, when these conditions are satisfied, the space $\lambda_b(B)$ is simply connected by arcs, simply connected, Galois with group $\pi_1(B, b)^{\circ}$, and the pointed covering $(\lambda_b(B), \varepsilon_b)$ is a universal covering of the pointed space $(B, b)$.

It follows from the definition of an unlaceable space that the subgroup of $\pi_1(B, b)$ reduced to the identity element is admissible if and only if B is unlaceable. This proves the equivalence (i)$\Leftrightarrow ($iv). Moreover, the equivalence of properties (iv) and (v) follows from Remarks 4 and 5 of III, p. 320.

(iv)$\Rightarrow ($ii). By III, p. 316, Prop. 5, there exists a covering $(E, p)$ of B and a point $x\in E_b$ such that $p_*(\pi_1(E, x)) =\{e\}$; in particular, $\pi_1(E, x)$ is reduced to the identity element. The connected component by arcs of $x$ in E is then a nonempty covering of B (I, p. 80, Cor. 1 to Prop. 6), simply connected by arcs.

(ii)$\Rightarrow ($iii). Let E be a nonempty covering of B which is simply connected by arcs. Let $x$ be a point of $E_b$ (there exists one because B is connected, I, p. 74, Prop. 4). The projection $q: E\rightarrow$ B induces a homeomorphism $\Lambda_x(E)\rightarrow \Lambda_b(B)$ (III, p. 302, Cor. 2 to Prop. 3), whence, by passing to connected components by arcs, a homeomorphism $q_*:\lambda_x(E)\simeq \lambda_b(B)$. The endpoint mapping $\Lambda_x(E)\rightarrow E$ is continuous and open, because E is locally connected by arcs (III, p. 264, Corollary). The mapping $\varepsilon_E:\lambda_x(E)\rightarrow E$ which it defines by passing to connected components by arcs is therefore continuous and open. The mapping $\varepsilon_E\circ (q_*)^{-1}:\lambda_b(B)\rightarrow E$ is bijective, continuous and open. This proves that the topological space $\lambda_b(B)$, endowed with the compact-convergence topology and the endpoint mapping, is a covering of B.

(iii)$\Rightarrow ($v). In fact, the set $\pi_1(B, b)$, endowed with the quotient topology of the compact-convergence topology, is identified with the fibre of the B-space $\lambda_b(B)$ over the class of the constant loop at $b$. If $\lambda_b(B)$ is a covering of B, then $\pi_1(B, b)$ is discrete for the compact-convergence topology.

Suppose that these assertions are verified. From the foregoing, the space $\lambda_b(B)$ is then a covering of B which is simply connected by arcs, hence simply connected. The pointed space $(\lambda_b(B), \varepsilon_b)$ is consequently a universal covering of $(B, b)$ (I, p. 126, corollary to Prop. 3) and the covering $\lambda_b(B)$ is a Galois covering of B (I, p. 120, Prop. 1). The canonical homomorphism $h_{(\lambda_b(B),\varepsilon_b)}:\pi_1(B, b)\rightarrow$ Aut$_B(\lambda_b(B))$ is then an isomorphism (III, p. 306, Prop. 5).