Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.
Affine geometry is a branch of geometry that studies the properties of figures that remain invariant under affine transformations, which include translation, scaling, and shearing. Unlike Euclidean geometry, affine geometry does not involve the concept of angle or distance, focusing instead on parallelism and ratios of lengths along parallel lines.
A 3-manifold is a space that locally resembles Euclidean 3-dimensional space, meaning each point has a neighborhood that looks like the Euclidean space R^3. Understanding 3-manifolds is crucial in topology and geometry, particularly in the study of the universe's shape and the field of 3-dimensional topology, where they serve as the primary objects of study.
Euler's Characteristic is a topological invariant that represents the relationship between the number of vertices, edges, and faces of a polyhedron. It is given by the formula V - E + F = χ, where χ is often 2 for convex polyhedra, highlighting foundational properties of geometric structures.