Connectedness

A topological space is said to be connected if it cannot be divided into two disjoint nonempty open sets whose union is the entire space.

The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f : [0, 1] -> X with f(0) = x and f(1) = y.

The connected (or path-connected) subsets of the real numbers R are called intervals.

Every path-connected space is connected. An example of a connected space that is not path-connected is the topologist's sine curve. This is the compact plane set

{ (x, y) in R² | 0 < x ≤ 1,  y = sin(1/x) } U { (0, y) in R² | -1 ≤ y ≤ 1 }.