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 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
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).
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.
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.