Lemma 1

Let B, $B'$ be topological spaces, and let $f: B'\rightarrow B$ be a continuous mapping. If the fibred square $B'\times_BB'$ is locally connected, the topological space $B'$ is locally connected.

Let $a$ be a point of $B'$ and let V be a neighbourhood of $a$. Suppose that the fibre square $B'\times_BB'$ is locally connected and let W be a connected neighbourhood of $(a, a)$ in $B'\times_BB'$ which is contained in $V\times V$. Put U = pr$_1(W)$. The set U is contained in V, and is connected since the image of a connected set under a continuous mapping is connected. If $\Delta_{B'}$ denotes the diagonal of $B'\times_BB'$, the mapping pr$_1|\Delta_{B'}: \Delta_{B'}\rightarrow B'$ is a homeomorphism. Since U contains pr$_1(W\cap \Delta_{B'})$ and $W\cap \Delta_{B'}$ is a neighbourhood of $(a, a)$ in $\Delta_{B'}$, U is a neighbourhood of $a$ in $B'$. This proves the lemma.