Arc (topology)

simple arc, Jordan arc

A topological space homeomorphic to the unit interval $[0,1]$. This should not be confused with a path: A continuous function with domain the unit interval.

An intrinsic characterization is: A simple arc is a line (curve) that has ramification index 1 at two points (the end points) and ramification index 2 at all other points (the interior points).

An arc in the plane is often defined by specifying the coordinates of its points as continuous functions $x=\phi(t)$, $y=\psi(t)$ of some parameter $t$, $a\leq t\leq b$; it is assumed in so doing that different values of $t$ correspond to different points.

A related concept is that of a Hausdorff arc (or sometimes also arc): A continuum (a compact and connected topological space) that is irreducibly connected between two of its points (any closed and connected subset containing these two points must be equal to the whole space).

References

[Ku]	K. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. Chapt. III, §3 (Translated from Polish)
[Ku2]	K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)