Path-connected space

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This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces

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

This article is about a basic definition in topology.
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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

Stronger properties

Weaker properties



This property of topological spaces is closed under taking arbitrary products
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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).


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).

Template:Connected union-closed

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