Difference between revisions of "Path-connected space"

From Topospaces
Jump to: navigation, search
(Relation with other properties)
Line 17: Line 17:
 
==Relation with other properties==
 
==Relation with other properties==
  
 +
==Stronger properties==
 +
 +
* [[Simply connected space]]
 +
* [[Contractible space]]
 
===Weaker properties===
 
===Weaker properties===
  

Revision as of 00:59, 27 October 2007

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 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

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.