Proposition 2

a) Every covering of an unloopable space is unloopable.

b) Conversely, let $p: E\rightarrow B$ be an étale and surjective mapping and suppose that E is an unloopable space. Then B is unloopable.

a) Let B be an unloopable topological space and let $(E, p)$ be a covering of B. Then the space E is locally arcwise connected. Let $x$ be a point of E, put $b=p(x)$, and let V be a neighbourhood of $b$ in B such that the canonical homomorphism $\pi_1(V, b)\rightarrow \pi_1(B, b)$ is trivial. Let $i: V\rightarrow B$ and $j:\overset{-1}{p}(V)\rightarrow E$ denote the canonical injections. By hypothesis, the image of the homomorphism $\pi_1(i, b)$ is reduced to the identity element. Since $p\circ i=j\circ p$ and $\pi_1(p, x)$ is injective, the image of the homomorphism $\pi_1(j, x)$ is reduced to the identity element. This proves that E is unloopable.

b) The space B is locally arcwise connected. Let $b$ be a point of B and let $x$ be a point of $E_b$. Since $p$ is étale, there exists a neighbourhood U of $x$ in E such that $p$ induces a homeomorphism of U onto $p(U)$. Let V be a neighbourhood of $x$ in E contained in U such that the homomorphism $\pi_1(j, x)$ is trivial, where $j: V\rightarrow E$ denotes the canonical injection. Let $q: V\rightarrow p(V)$ be the mapping deduced from $p$ by passing to subspaces; it is a homeomorphism. Let also $i$ denote the canonical injection of $p(V)$ into B. We have $p\circ j=i\circ q$, hence $\pi_1(i, b)\circ \pi_1(q, x) =\pi_1(p, x)\circ \pi_1(j, x)$ is the trivial homomorphism. Since the homomorphism $\pi_1(q, x)$ is an isomorphism, the homomorphism $\pi_1(i, b)$ is trivial. It follows that the space B is unloopable.