# Path-connected space

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** 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

### 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.