In topology, continuity refers to the idea that a function is able to preserve the inherent structure of its domain. More specifically, a function is continuous if, for every open set in its range, the inverse image of that open set is also open in the domain.

Intuitively, this means that a continuous function is able to "stretch" or "bend" the points in its domain without breaking or "tearing" the inherent structure of that domain. This property is important because it allows us to make meaningful topological statements about the relationships between different spaces and the functions that map between them.

For example, consider a function that maps points on a sphere to points on a plane. If this function is continuous, it means that any open set on the plane (such as a circle or a line segment) will be mapped back to an open set on the sphere (such as a "slice" of the sphere or a loop around the equator). This allows us to say that, topologically, the sphere and the plane are "similar" in some sense, because the function is able to preserve the structure of one space while mapping it to the other.