A topological space is path-connected if for every pair of points in the space, there is a continuous path connecting them. This means that it is possible to get from any point in the space to any other point by following a continuous curve.

For example, consider the real numbers with the standard topology (where open sets are defined as unions of intervals). The real numbers are path-connected, because for any two points x and y, there is a continuous path connecting them, namely the straight line segment from x to y.

On the other hand, the unit circle in the plane is not path-connected, because there is no continuous path connecting any point on the circle to the point diametrically opposite to it.

Path-connectedness is an important concept in topology, because it gives us a way to define the "shape" of a topological space. For example, a space that is path-connected is often thought of as "connected" in some sense, while a space that is not path-connected may be thought of as "disconnected" or "separated" into different pieces.