Difference between revisions of "Path-connected space"
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{{variationof|connectedness}} | {{variationof|connectedness}} | ||
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==Definition== | ==Definition== | ||
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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== | ||
+ | ===Stronger properties=== | ||
+ | |||
+ | * [[Simply connected space]] | ||
+ | * [[Contractible space]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
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==Metaproperties== | ==Metaproperties== | ||
− | { | + | {| class="sortable" border="1" |
− | + | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | |
− | A | + | |- |
− | + | | [[dissatisfies metaproperty::subspace-hereditary property of topological spaces]] || No || [[path-connectedness is not hereditary]] || It is possible to have a path-connected space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is not path-connected in the subspace topology. | |
− | { | + | |- |
− | + | | [[dissatisfies metaproperty::weakly hereditary property of topological spaces]] || No || [[path-connectedness is not weakly hereditary]] || It is possible to have a path-connected space <math>X</math> and a closed subset <math>A</math> of <math>X</math> such that <math>A</math> is not path-connected in the subspace topology. | |
− | + | |- | |
− | + | | [[satisfies metaproperty::product-closed property of topological spaces]]|| Yes || [[path-connectedness is product-closed]] || Suppose <math>X_i, i \in I</math>, are all path-connected spaces. Then, the Cartesian product <math>\prod_{i \in I} X_i</math> is also a path-connected space with the [[product topology]]. | |
− | + | |- | |
+ | | [[dissatisfies metaproperty::box product-closed property of topological spaces]] || No || [[path-connectedness is not box product-closed]] || It is possible to have <math>X_i, i \in I</math> all path-connected spaces such that the Cartesian product <math>\prod_{i \in I} X_i</math> is ''not'' path-connected in the [[box topology]]. | ||
+ | |- | ||
+ | | [[satisfies metaproperty::coarsening-preserved property of topological spaces]] || Yes || [[path-connectedness is coarsening-preserved]] || If <math>X</math> is path-connected under a topology <math>\tau</math>, it remains path-connected when we pass to a [[coarser topology]] than <math>\tau</math>. | ||
+ | |- | ||
+ | | [[satisfies metaproperty::continuous image-closed property of topological spaces]] || Yes || [[path-connectedness is continuous image-closed]] || If <math>X</math> is a path-connected space and <math>Y</math> is the image of <math>X</math> under a continuous map, then <math>Y</math> is also path-connected. | ||
+ | |- | ||
+ | | [[satisfies metaproperty::connected union-closed property of topological spaces]] || Yes || [[path-connectedness is connected union-closed]] || | ||
+ | |- | ||
+ | | [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[path-connectedness is not closure-preserved]] || It is possible to have a <math>A</math> a subset of <math>X</math> that is path-connected in the subspace topology but such that the closure <math>\overline{A}</math> is not path-connected in its subspace topology. | ||
+ | |} | ||
− | + | ==References== | |
+ | ===Textbook references=== | ||
+ | * {{booklink-defined|Munkres}}, Page 155 (formal definition) | ||
+ | * {{booklink-defined|SingerThorpe}}, Page 52 (formal definition): introduced under name '''arcwise connected space''' | ||
+ | * {{booklink-defined|Rotman}}, Page 25 (formal definition) |
Latest revision as of 19:05, 26 January 2012
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.
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
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)