# 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

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## 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 $\gamma(1) = b$.

## 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. MunkresMore info, Page 155 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. ThorpeMore info, Page 52 (formal definition): introduced under name arcwise connected space