Example 2

Let Y be a topological space locally connected by arcs, and let X be an open subset of Y. The space $\mathscr{C}_c(\mathbf{I}; X)$ of paths in X is identified with a subspace of the space $\mathscr{C}_c(\mathbf{I}; Y)$ of paths in Y; suppose that it is dense. The canonical injection of X into Y is then a relatively connected mapping.

It is enough, in fact, to prove that, for every connected and nonempty open subset V of Y, the set $V\cap X$ is connected and nonempty. The space $\mathscr{C}_c(\mathbf{I}; V)$ is a nonempty open subset of $\mathscr{C}_c(\mathbf{I}; Y)$; it meets $\mathscr{C}_c(\mathbf{I}; X)$, which proves that $V\cap X=\not\emptyset$. Let $x$ and $x'$ be points of $V\cap X$. Since $V\cap X$ is locally connected by arcs, the points $x$ and $x'$ have open neighbourhoods U and $U'$, connected by arcs and contained in $V\cap X$. Since V is connected by arcs (III, p. 260, prop. 7), there exists a path in V joining $x$ to $x'$. By the density assumption and the definition of the topology of compact convergence, there exists a path in $V\cap X$ joining a point of U to a point of $U'$. There then exists a path joining $x$ to $x'$ in $V\cap X$, which proves that the set $V\cap X$ is connected.