Difference between revisions of "Path-connected space"

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(Relation with other properties)
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* [[Connected space]]: {{proofat|[[Path-connected implies connected]]}}
 
* [[Connected space]]: {{proofat|[[Path-connected implies connected]]}}
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==Metaproperties==
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{{DP-closed}}
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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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{{connected union-closed}}
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A union of a family of path-connected subsets having nonempty intersection, is path-connected.

Revision as of 04:58, 18 August 2007

This is a variation of connectedness. View other variations of connectedness

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 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

Weaker properties

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).

Template:Connected union-closed

A union of a family of path-connected subsets having nonempty intersection, is path-connected.