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Let
![$ I=[0,1] \subset \mathbb{R}$](http://images.planetmath.org:8080/cache/objects/942/l2h/img1.png "$ I=[0,1] \subset \mathbb{R}$") 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.
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"path" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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See Also: simple path, distance (in a graph), locally connected, example of a connected space that is not path-connected, (path) connectness as a homotopy invariant
| Other names: |
pathwise connected, path-connected, path connected |
| Also defines: |
path, arc, arcwise connected, initial point, terminal point |
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Cross-references: iff, Hausdorff spaces, connected space, analytic, rectifiable paths, piecewise, differentiable, classes, structure, homeomorphism, reparameterization, equivalent, one-to-one, map, addition, point, continuous map, topological space
There are 138 references to this entry.
This is version 11 of path, born on 2001-11-17, modified 2006-12-13.
Object id is 942, canonical name is PathConnected.
Accessed 17545 times total.
Classification:
| AMS MSC: | 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces ) |
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Pending Errata and Addenda
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