Corollary 2

Let B be an unwindable topological space and let $b$ be a point of B. Let $(E, x)$ be a universal covering of the pointed space $(B, b)$. For every covering $E'$ of B and every point $x'$ of $E'_b$, the unique B-morphism $f: E\rightarrow E'$ such that $f(x) =x'$ makes E a covering of $E'$. The pointed space $(E, x)$ is then a universal covering of $(E', x')$.

By prop. 7 of I, p. 81, the space E, endowed with the mapping $f$, is a covering of $E'$. The last assertion then follows from Corollary 1.

Let B be an unwindable topological space and let $a,b$ be two points of B. The quotient topological space $\varpi_{a,b}(B)$ is homeomorphic to the space $\pi_1(B, a)$, endowed with the quotient topology of the compact-open topology (III, p. 293, remark 3), and therefore is discrete by Theorem 1 of IV, p. 342. Consequently, the strict homotopy class of every path in B joining $a$ to $b$ is an open subset of $\Lambda_{a,b}(B)$. More generally: