Difference between revisions of "Path-connected space"
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===Symbol-free 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. | + | 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=== | ===Definition with symbols=== | ||
− | A [[topological space]] <math>X</math> is said to be '''path-connected''' if for any two points <math>a,b \in X</math> there is a continuous map <math>\gamma:[0,1] \to X</math> such that <math>\gamma(0) = a</math> and <math>\gamma(1) = b</math>. | + | A [[topological space]] <math>X</math> is said to be '''path-connected''' or '''arc-wise connected''' if for any two points <math>a,b \in X</math> there is a continuous map <math>\gamma:[0,1] \to X</math> such that <math>\gamma(0) = a</math> and <math>\gamma(1) = b</math>. |
==Relation with other properties== | ==Relation with other properties== | ||
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==References== | ==References== | ||
===Textbook references=== | ===Textbook references=== | ||
− | * {{booklink|Munkres}}, Page 155 (formal definition) | + | * {{booklink-defined|Munkres}}, Page 155 (formal definition) |
− | * {{booklink|SingerThorpe}}, Page 52 (formal definition): introduced under name '''arcwise connected space''' | + | * {{booklink-defined|SingerThorpe}}, Page 52 (formal definition): introduced under name '''arcwise connected space''' |
− | * {{booklink|Rotman}}, Page 25 (formal definition) | + | * {{booklink-defined|Rotman}}, Page 25 (formal definition) |
Revision as of 20:10, 20 July 2008
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
Contents |
This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology
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
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
A direct product of path-connected spaces is path-connected. This is true both for finite and infinite direct products (using the product topology for infinite direct products).
Coarsening
This property of topological spaces is preserved under coarsening, viz, if a set with a given topology has the property, the same set with a coarser topology also has the property
Shifting to a coarser topology preserves the property of being path-connected. This is because a path in a finer topology continues to remain a path in a coarser topology -- we simply compose with the identity map from the finer to the coarser topology (which, by definition, must be continuous).
Template:Connected union-closed
A union of a family of path-connected subsets having nonempty intersection, is path-connected.
Retract-hereditariness
This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces
Any retract of a path-connected space is path-connected.
Closure under continuous images
The image, via a continuous map, of a topological space having this property, also has this property
More generally, the image of a path-connected set under a continuous map is again path-connected.
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)
- Homotopy-invariant properties of topological spaces
- Properties of topological spaces
- Basic definitions in topology
- Standard terminology
- Coarsening-preserved properties of topological spaces
- Retract-hereditary properties of topological spaces
- Properties of topological spaces closed under taking continuous images