mathematical topology

Mathematical topology is the study of properties of shapes and spaces that remain unchanged under continuous deformations, like stretching or bending, but not tearing or gluing. It focuses on the intrinsic nature of spaces, such as whether they are connected or have holes, rather than exact measurements. For example, a coffee mug and a donut are topologically equivalent because each has one hole, despite their different shapes. Topology helps us understand the fundamental structure of complex spaces and how they can be transformed without losing their essential features.