Which branch of mathematics studies properties unchanged by stretching and bending?

Topology looks at what stays the same about a shape when you deform it continuously, allowing stretching, squashing and twisting but never cutting or gluing. Distance and angle are irrelevant there, so a square and a circle are the same object, and the famous line is that a topologist cannot tell a coffee mug from a doughnut, since each has exactly one hole and one can be moulded into the other. What survives such deformation are properties like the number of holes, whether a space is connected, and whether a surface has two sides, which is why the one-sided Mobius strip is a standard example. The field grew out of problems such as the bridges of Konigsberg, where Euler showed that only the pattern of connections mattered, not the map's geometry. Topology now underpins large parts of modern mathematics and appears in physics, data analysis and the study of networks.

Euler's formula relating vertices, edges and faces of a solid is an early topological result, since it holds however the shape is deformed.