Path-connected space
Definition
A topological space is said to be path-connected if there is a path connecting any two points in the space. More formally, a topological space X is path-connected if for every pair of points x, y in X, there exists a continuous function f : [0,1] → X such that f(0) = x and f(1) = y. This function f is known as a path from x to y.
Properties
Path-connectedness is a stronger condition than connectedness. Every path-connected space is connected, but the converse is not always true. For example, the Topologist's Sine Curve is a connected space that is not path-connected.
A subset of a topological space is path-connected if and only if it is path-connected in the subspace topology. This is not true for connectedness in general.
The continuous image of a path-connected space is path-connected. This follows from the fact that the composition of continuous functions is continuous.
The product of path-connected spaces, endowed with the product topology, is path-connected. This is a consequence of the fact that a product of paths is a path.
Examples
1. The real numbers R with the standard topology are path-connected. Given any two points x, y in R, the function f(t) = (1-t)x + ty is a path from x to y.
2. The unit disk in R², including its boundary, is path-connected. Given any two points x, y in the disk, the straight line segment connecting x and y is a path from x to y.
3. The unit circle in R², excluding the origin, is path-connected. Given any two points x, y on the circle, the arc of the circle connecting x and y is a path from x to y.
Path-Connectedness in Metric Spaces
In a metric space, the concept of path-connectedness can be related to the concept of metric. A metric space is path-connected if and only if for any two points x, y in the space, there exists a path of finite length connecting x and y.
Path-Connectedness in Complex Analysis
In complex analysis, path-connectedness plays a crucial role in the definition of the complex integral. The complex integral of a function along a path depends only on the endpoints of the path, not on the specific path taken, as long as the function is holomorphic on a path-connected set containing the path.