Let and let be a topological space.
A continuous map such that and is called a path in . The point is called the initial point of the path and is called its terminal point. If, in addition, the map is one-to-one, then it is known as an arc.
Sometimes, it is convenient to regard two paths or arcs as equivalent if they differ by a reparameterization. That is to say, we regard and as equivalent if there exists a homeomorphism such that and and .
If the space has extra structure, one may choose to restrict the classes of paths and reparameterizations. For example, if has a differentiable structure, one may consider the class of differentiable paths. Likewise, one can speak of piecewise linear paths, rectifiable paths, and analytic paths in suitable contexts.
The space is said to be pathwise connected if, for every two points , there exists a path having as initial point and as terminal point. Likewise, the space is said to be arcwise connected if, for every two distinct points , there exists an arc having as initial point and as terminal point.
A pathwise connected space is always a connected space, but a connected space need not be path connected. An arcwise connected space is always a pathwise connected space, but a pathwise connected space need not be arcwise connected. As it turns out, for Hausdorff spaces these two notions coincide — a Hausdorff space is pathwise connected iff it is arcwise connected.
|Date of creation||2013-03-22 12:00:15|
|Last modified on||2013-03-22 12:00:15|
|Last modified by||rspuzio (6075)|