Proposition 11

A connected and locally connected topological space, whose topology can be defined by a distance for which it is complete, is connected by arcs.

Let X be a topological space. Call a train in X any nonempty finite sequence $T = (W_i)_{1\leqslant i\leqslant n}$ of connected open subsets of X such that $W_i\cap W_{i+1}=\not\emptyset$ for $1\leqslant i\leqslant n-1$. We say that $n$ is the length of the train T and that the $W_i$ are its wagons. If X is endowed with a distance compatible with its topology, we call the width of the train T the maximum of the diameters of its wagons. We say that the train joins a point $a$ to a point $b$ if $a$ belongs to the first and $b$ to the last wagon. We call a refinement of T any pair $(T', f)$ formed by a train $T'= (W'_j)_{1\leqslant j\leqslant m}$ and a strictly increasing mapping $f:\{0,1, . . . , n\} \rightarrow \{0,1, . . . , m\}$ such that $f(0) = 0,f(n) =m$ and $W'_j\subset W_i$ for $1\leqslant i\leqslant n$ and $f(i-1)< j\leqslant f(i)$.