Path-connected space
This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spacesView other homotopy-invariant properties of topological spaces OR view all properties of topological spaces
This is a variation of connectedness. View other variations of connectedness
This article is about a basic definition in topology.
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Contents
Definition
Symbol-free definition
A topological space is said to be path-connected or arc-wise connected if given any two points on the topological space, there is a path (or an arc) starting at one point and ending at the other.
Definition with symbols
A topological space is said to be path-connected or arc-wise connected if for any two points there is a continuous map such that and .
Relation with other properties
Stronger properties
Weaker properties
- Connected space: For full proof, refer: Path-connected implies connected
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subspace-hereditary property of topological spaces | No | path-connectedness is not hereditary | It is possible to have a path-connected space and a subset of such that is not path-connected in the subspace topology. |
weakly hereditary property of topological spaces | No | path-connectedness is not weakly hereditary | It is possible to have a path-connected space and a closed subset of such that is not path-connected in the subspace topology. |
product-closed property of topological spaces | Yes | path-connectedness is product-closed | Suppose , are all path-connected spaces. Then, the Cartesian product is also a path-connected space with the product topology. |
box product-closed property of topological spaces | No | path-connectedness is not box product-closed | It is possible to have all path-connected spaces such that the Cartesian product is not path-connected in the box topology. |
coarsening-preserved property of topological spaces | Yes | path-connectedness is coarsening-preserved | If is path-connected under a topology , it remains path-connected when we pass to a coarser topology than . |
continuous image-closed property of topological spaces | Yes | path-connectedness is continuous image-closed | If is a path-connected space and is the image of under a continuous map, then is also path-connected. |
connected union-closed property of topological spaces | Yes | path-connectedness is connected union-closed | |
closure-preserved property of topological spaces | No | path-connectedness is not closure-preserved | It is possible to have a a subset of that is path-connected in the subspace topology but such that the closure is not path-connected in its subspace topology. |
References
Textbook references
- Topology (2nd edition) by James R. Munkres^{More info}, Page 155 (formal definition)
- Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 52 (formal definition): introduced under name arcwise connected space
- An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman^{More info}, Page 25 (formal definition)