Topological Manifolds

A topological manifold is a mathematical space that, on a small scale, resembles everyday Euclidean space (like a line, plane, or three-dimensional space). This means that around every point, you can find a neighborhood that can be smoothly stretched or bent into a standard shape without tearing or gluing. These spaces can be oddly shaped or curved globally but look "flat" locally. Examples include the surface of a sphere or a doughnut-shaped object, where each small piece behaves like a flat sheet, enabling calculus and other mathematical analysis to be performed.