Intersections of lines, for example, remain intersections, and the hole in a torus {doughnut} cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup {the hole turning into a handle} but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or "invariant," patterns.
In topology, certain shapes maintain their fundamental characteristics despite transformations. For instance, the points where lines cross will always intersect, and a torus, like a doughnut, has a permanent hole. This principle allows a torus to be reshaped into a coffee cup, where the hole becomes the handle, but it cannot be reshaped into something like a pancake since this would alter its core properties.
Topology focuses on the relationships between objects and emphasizes aspects that remain invariant under continuous transformations. It represents a mathematical exploration of how certain features are preserved, highlighting the importance of understanding patterns that do not change, regardless of the form an object takes.
