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But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used. The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and it is a manifold, but not a closed manifold.
An open manifold is a manifold without boundary (not necessarily connected), with no compact component. For instance, [,] is a compact manifold, is a closed manifold, and (,) is an open manifold, while [,) is none of these.
Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never countable , unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola , a hyperbola , and the locus of points on a cubic curve y 2 = x 3 − x (a closed loop piece and an open ...
A closed surface is a surface that is compact and without boundary. Examples of closed surfaces include the sphere, the torus and the Klein bottle. Examples of non-closed surfaces include an open disk (which is a sphere with a puncture), a cylinder (which is a sphere with two punctures), and the Möbius strip.
Hydraulic manifold A component used to regulate fluid flow in a hydraulic system, thus controlling the transfer of power between actuators and pumps Inlet manifold (or "intake manifold") An engine part that supplies the air or fuel/air mixture to the cylinders Scuba manifold In a scuba set, connects two or more diving cylinders Vacuum gas manifold
Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, it is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R 3, the Klein bottle cannot.
A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the Friedmann–Lemaître–Robertson–Walker (FLRW) model, the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.
The prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds. A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension.