Topology: Concepts, History, and Applications

Topological invariance and continuous deformation

Topological invariance concerns properties of geometric objects that remain unchanged under continuous deformations such as stretching, twisting, bending, or crumpling, provided the object is not cut, glued, punctured, or passed through itself. These preserved properties are called topological properties and include dimension, compactness, connectedness, orientability, and the number of holes. Thus, a circle and a square are topologically equivalent, while a sphere and a torus are not because the torus has a hole. The main notion of continuous deformation is homeomorphism: two spaces are homeomorphic when there is a continuous, one-to-one correspondence between them whose inverse is also continuous. This explains why a coffee mug and a doughnut are considered topologically identical, since each has one hole and can be continuously reshaped into the other. A broader relation, homotopy equivalence, allows spaces to be continuously squashed or expanded into one another and is central to algebraic topology. These ideas enable mathematicians to study essential structural features independently of an object's precise geometric shape.

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Topological spaces and topologies

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