pathconnectedness

Pathconnectedness, or path connectivity, is a property of a topological space X defined as follows: for any two points x and y in X there exists a continuous function f from the closed unit interval [0,1] to X with f(0)=x and f(1)=y. Such a function f is called a path from x to y.

From this notion comes the concept of path components: the maximal path-connected subspaces of X; every point lies in a unique path component, and X is the disjoint union of its path components. In general, a path component is contained in the corresponding connected component. If the space is locally path-connected, then the path components and the connected components coincide.

Examples help illustrate the idea. The Euclidean spaces R^n and the circle S^1 are path-connected. The rationals Q with the usual topology are totally disconnected (not path-connected). The topologist's sine curve is connected but not path-connected. The product of two path-connected spaces is path-connected, and the image of a path-connected space under a continuous map is path-connected.

Relation to other concepts: if X is path-connected, the fundamental group can be defined at any basepoint and any two basepoints yield isomorphic groups (via paths between basepoints). Path-connectedness is preserved under homeomorphisms and under continuous maps, and it is a basic topological property that reflects how a space can be traversed by continuous paths.