Proposition 18

Let X be a separated topological space. Let $a$ and $b$ be distinct points of X which belong to the same arcwise connected component of X. There exists an injective path joining $a$ to $b$ in X.

Let $f:\mathbf{I}\rightarrow X$ be a path joining $a$ to $b$ in X. Let $\mathscr{U}$ be the set of open subsets U of $]0,1[$ such that $f(x) =f(y)$ for every connected component $]x, y[$ of U.