# Difference between revisions of "Path-connected space"

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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

## Contents

## 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** if for any two points there is a continuous map such that and .

## Relation with other 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).

Template:Connected union-closed

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