TOPICS

# Manifold

A manifold is a topological space that is locally Euclidean, i.e., around every point, there is a neighborhood that is topologically the same as an open unit ball in some dimension.

Manifold is a graduate-level concept that would be first encountered in a topology course.

### Examples

 Euclidean Space: Euclidean space of dimension n is the space of all n-tuples of real numbers which generalizes the two-dimensional plane and three-dimensional space. Möbius Strip: A Moebius strip is one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving it half a twist, and then reattaching the two ends. Projective Plane: The projective plane is the set of lines in the Euclidean plane that pass through the origin. It can also be viewed as the Euclidean plane together with a line at infinity. Sphere: A sphere is the set of all points in three-dimensional space that are located at a fixed distance from a given point. Torus: A torus is a closed surface containing a single hole that is shaped like a doughnut.

### Prerequisites

 Topological Space: A topological space is a set with a collection of subsets T that together satisfy a certain set of axioms defining the topology of that set.