Proposition 1

The product space of a finite family of unloopable spaces is unloopable.

It is sufficient to prove that, if A and B are two unloopable spaces, the same is true of their product $A\times B$. Under these conditions, the space $A\times B$ is in fact locally arcwise connected (III, p. 261, prop. 9). Let $(a, b)$ be a point of $A\times B$. By hypothesis there exists a neighbourhood U of $a$ (resp. a neighbourhood V of $b$) such that the image of the homomorphism $i_*:\pi_1(U, a)\rightarrow \pi_1(A, a)$ deduced from the canonical injection $i: U\rightarrow A$ (resp. of the homomorphism $j_*:\pi_1(V, b)\rightarrow \pi_1(B, b)$ deduced from the canonical injection $j: V\rightarrow B$) is reduced to the identity element. Then $U\times V$ is a neighbourhood of $(a, b)$ in $A\times B$. The homomorphism $\pi_1(U\times V,(a, b))\rightarrow \pi_1(A\times B,(a, b))$ identifies with the homomorphism $(i_*, j_*)$ (III, p. 297, corollary of prop. 4). Its image is therefore reduced to the identity element. This proves that $A\times B$ is unloopable.