Path-connected space

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

• 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).

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