Path-connected space

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 or arc-wise 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

• Simply connected space
• Contractible space

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).

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

Retract-hereditariness

This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces

Any retract of a path-connected space is path-connected.

Closure under continuous images

The image, via a continuous map, of a topological space having this property, also has this property

More generally, the image of a path-connected set under a continuous map is again path-connected.

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 155 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 52 (formal definition): introduced under name arcwise connected space
• An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman^{More info}, Page 25 (formal definition)