Arc

A topological space $(X,\tau)$ is called an arc if it is homeomorphic to the real interval $[0,1]$. If $h \colon [0,1] \rightarrow X$ is a homeomorphism onto the arc $X$ then the end points of the arc $X$ are the points $h(0)$ and $h(1)$.

Properties of arcs

Theorem: If $X$ is an arc then $X$ is a continuum.

Proof: To prove this theorem is suffices to show that all arcs are compact and connected. █

Onto monotone map of an interval is an arc