topology of the plane

The topology of the plane studies properties that remain unchanged when the plane is stretched or deformed without tearing or gluing. Think of it as focusing on shape features like connectedness and continuity, rather than exact distances or angles. For example, a circle and an ellipse are topologically equivalent because one can be smoothly transformed into the other without cutting. Topology helps us understand how spaces are connected and how they can be deformed, revealing fundamental, flexible aspects of shapes and their relationships in the plane.