pathconnected

Path connectedness is a property of topological spaces that describes the ability to connect any two points within the space via continuous paths. Formally, a topological space is called path connected if for any two points in the space, there exists a continuous function, called a path, mapping from the closed interval [0, 1] into the space such that the path starts at the first point and ends at the second. In other words, for points x and y in the space, there is a continuous function f: [0, 1] → X with f(0) = x and f(1) = y.

Path connectedness is a stronger condition than connectedness, which only requires the space to be indivisible into two disjoint non-empty open sets. A path connected space implies connectedness, but the converse is not necessarily true. For example, the topologist’s sine curve is connected but not path connected.

Path connectedness is an important concept in various fields of mathematics, including algebraic topology, where it influences the classification of spaces by their fundamental groups and related invariants. It also plays a role in continuum theory and the study of metric spaces.

The property is invariant under homeomorphisms, meaning that spaces that are topologically equivalent share the same path connectedness status. Many familiar shapes, such as lines, polygons, disks, and spheres, are path connected, whereas spaces with certain topological complexities may lack this property.

Understanding path connectedness helps in analyzing the structure of spaces, especially regarding continuous deformations and the existence of paths within them.