In topology, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics.

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.

6 days ago · A manifold may be endowed with more structure than a locally Euclidean topology. For example, it could be smooth , complex , or even algebraic (in order of specificity). A smooth manifold with a metric is called a Riemannian manifold , and one with a symplectic structure is called a symplectic manifold .

Loosely manifolds are topological spaces that look locally like Euclidean space. A little more precisely it is a space together with a way of identifying it locally with a Euclidean space which is compatible on overlaps.

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3- manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer.

All 3 manifolds in this chapter are assumed to be connected, orientable, and compact, possibly with boundary, unless otherwise stated or constructed. 1. Prime Decomposition. Implicit in the prime decomposition theorem is the fact that S3 is prime, other-wise one could only hope for a prime decomposition modulo invertible elements, as in algebra.

Jun 11, 2024 · A topological manifold is a topological space (usually required to be Hausdorff and paracompact) which is locally homeomorphic to a Euclidean space ℝ n \mathbb{R}^n equipped with its metric topology.

Mar 31, 2025 · A topological space satisfying some separability (i.e., it is a T2-space) and countability (i.e., it is a paracompact space) conditions such that every point has a neighborhood homeomorphic to an open set in for some . Every smooth manifold is a topological manifold, but not necessarily vice versa.

An n-dimensional topological manifold is a topological space that is Haus-dorff, has a countable basis at every point, and is locally Euclidean. That is, every point has a neighbourhood which is homeomorphic to an open set of Rn. An n-dimensional differentiable manifold M is an n-dimensional topolog-ical manifold with a differentiable ...

Jun 7, 2020 · The branch of the theory of manifolds (cf. Manifold) concerned with the study of relations between different types of manifolds. The most important types of finite-dimensional manifolds and relations between them are illustrated in (1).