A manifold is a topological space with the property that every point's neighbourhood is homeomorphic to Euclidean space's. Informally, this means that when you "zoom in" on a point, the space around said point will eventually become indistinguishable to a Euclidean space. Examples of basic types of manifold are listed below.
- Euclidean space
- Real projective space
- Complex projective space
- Quaternionic projective space
0-manifolds are created from disjoint unions of 1 or more points, and are classified by their cardinality. Examples of 0-manifolds include the point, the point pair (0D hypersphere), the point triplet, and the natural line (along with other lines constructed from countable sets).
1-manifolds are also known as curves, and they are created from deforming, stretching, or taking disjoint unions of circles or real lines. Examples of 1-manifolds (excluding the aforementioned) include the open interval, circle pair, ellipse, parabola, hyperbola, cubic curve, real projective line, and the long line.
2-manifolds are also known as surfaces. They can be (deformed) subsets of a Euclidean plane, but they do not have to be. Examples of 2-manifolds include the Euclidean plane, hyperbolic plane, open disc, sphere, torus, double torus, real projective plane, Klein bottle, tube, and the Möbius strip.