Difference between revisions of "Path-connected space"

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(Metaproperties)
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==Metaproperties==
 
==Metaproperties==
  
{{DP-closed}}
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{| class="sortable" border="1"
 
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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
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).
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| [[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.
{{coarsening-preserved}}
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| [[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.
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).
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| [[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]].
{{connected union-closed}}
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| [[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]].
A union of a family of path-connected subsets having nonempty intersection, is path-connected.
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| [[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>.
{{retract-hereditary}}
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| [[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.
Any retract of a path-connected space is path-connected.
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| [[satisfies metaproperty::connected union-closed property of topological spaces]] || Yes || [[path-connectedness is connected union-closed]] ||
{{continuous image-closed}}
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| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[path-connectedness is 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.
More generally, the image of a path-connected set under a continuous map is again path-connected.
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==References==
 
==References==

Revision as of 18:59, 26 January 2012

This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces


View 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 X is said to be path-connected or arc-wise connected if for any two points a,b \in X there is a continuous map \gamma:[0,1] \to X such that \gamma(0) = a and \gamma(1) = b.

Relation with other properties

Stronger properties

Weaker properties

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 X and a subset A of X such that A 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 X and a closed subset A of X such that A is not path-connected in the subspace topology.
product-closed property of topological spaces Yes path-connectedness is product-closed Suppose X_i, i \in I, are all path-connected spaces. Then, the Cartesian product \prod_{i \in I} X_i 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 X_i, i \in I all path-connected spaces such that the Cartesian product \prod_{i \in I} X_i is not path-connected in the box topology.
coarsening-preserved property of topological spaces Yes path-connectedness is coarsening-preserved If X is path-connected under a topology \tau, it remains path-connected when we pass to a coarser topology than \tau.
continuous image-closed property of topological spaces Yes path-connectedness is continuous image-closed If X is a path-connected space and Y is the image of X under a continuous map, then Y 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 closure-preserved It is possible to have a A a subset of X that is path-connected in the subspace topology but such that the closure \overline{A} is not path-connected in its subspace topology.

References

Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, Page 155 (formal definition)
  • Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. ThorpeMore info, Page 52 (formal definition): introduced under name arcwise connected space
  • An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. RotmanMore info, Page 25 (formal definition)