This is a variation of connectedness. View other variations of connectedness
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.