Arc (topology, $A\cong [0,1]$)

An arc is a space homeomorphic to $[0,1]$, or a subset with that property. A continuous injective map from $[0,1]$ into a Hausdorff space is a homeomorphism onto its image because its domain is compact. A line segment and a semicircle are arcs; a full circle is not. A continuous path may retrace points or cross itself, so its image need not be an arc.

An open arc is a topological space homeomorphic to an open interval. Unlike an arc, which is homeomorphic to a closed interval, an open arc does not include its two endpoints.