Proposition 15

The image of a path in a separated topological space is a compact, metrizable, connected and locally connected by arcs topological space.

Let X be a separated topological space and $c:\mathbf{I}\rightarrow X$ a continuous mapping. Let R denote the equivalence relation $c(s) =c(t)$ in $\mathbf{I}$. The topological space $c(\mathbf{I})$ is separated (TG, I, p. 63, cor. 1), the space $\mathbf{I}/R$ is quasi-compact (TG, I, p. 62, th. 2), hence the bijection $\mathbf{I}/R\rightarrow c(\mathbf{I})$ deduced from $c$ is a homeomorphism (TG, I, p. 63, cor. 2). Consequently, the space $c(\mathbf{I})$ is compact, metrizable (TG, IX, p. 22, prop. 17), connected (TG, I, p. 82, prop. 6) and locally connected by arcs (III, p. 261, prop. 8).