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Any covering space of a differentiable manifold may be equipped with a ( natural ) differentiable structure that turns p ( the covering map in question ) into a local diffeomorphism – a map with constant rank n.
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Any and covering
: Any form of removable court covering, including carpeting and artificial turf.
Any water in the glass can prevent excitement of the gas by covering designs set in the bottom of the glass, thus making the beer flat.
Any and space
* Any finite topological space, including the empty set, is compact.
* Any space carrying the cofinite topology is compact.
* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.
Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.
Any disturbance created in a sufficiently small region of isotropic space ( or in an isotropic medium ) propagates from that region in all radial directions.
Any major failure to a spacecraft en route is likely to be fatal, and even a minor one could have dangerous results if not repaired quickly, something difficult to accomplish in open space.
* Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra.
Any object gravitationally held by the rotating Earth – Moon system will be attracted to the barycenter to an equal and opposite degree as its tendency to fly off into space.
Any fire in a crew space will be automatically extinguished with Halon or another fire suppressant.
Any topological space which is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself.
Any second-countable space is separable: if is a countable base, choosing any gives a countable dense subset.
* Any continuous image of a separable space is separable.
* Any topological space which is the union of a countable number of separable subspaces is separable.
Any brain tumor is inherently serious and life-threatening because of its invasive and infiltrative character in the limited space of the intracranial cavity.
Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: a homothetic transformation, an isometry, and a special conformal transformation.
Any number of playing pieces may occupy the same space at the same time.
# Any phenomena that would seem attributable to absolute space and time ( e. g. inertia, and centrifugal force ) should instead be seen as emerging from the large scale distribution of matter in the universe.
* Any topological space can be viewed as a category by taking the open sets as objects, and a single morphism between two open sets and if and only if.
Any relatively flat, open space can be used as a terrain.
Any attempt to contact other worlds was forbidden, and the planetary government maintained an isolationist stance, forbidding space exploration of any kind.
Any competitor may rent office space in an incumbent's central office, place equipment to interconnect with their network, or purchase other related services.
* Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.
Any and differentiable
Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian.
* Any analytic function is smooth, that is, infinitely differentiable.
* Noether's theorem: Any quantity which has a continuous differentiable symmetry in the action has an associated conservation law.
1 ) Any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold with no differentiable structure ( so cannot be smoothed ).
Any and manifold
# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.
* Any smooth manifold is a diffeological space.
Any Calabi – Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi – Yau manifold.
Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology.
# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.
Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system.
Any vector bundle V over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle ; these universal Chern classes in turn can be explicitly written down in terms of Schubert cycles.
Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields.
* Any open manifold admits a ( non-complete ) Riemannian metric of positive ( or negative ) curvature.
Any “ nice ” space such as a manifold or CW complex is semi-locally simply connected.
Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.
* Any manifold with constant sectional curvature is an Einstein manifold — in particular:
Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.
Any manifold is an n-handlebody, that is, any manifold is the union of handles.
Any handlebody decomposition of a manifold defines a CW complex decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell.
Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot – Carathéodory, defined as
Any compact manifold has isoperimetric dimension 0.
* Any compact symplectic manifold admits a symplectic Lefschetz pencil ( Donaldson 1999 ).
Any symplectic manifold ( or indeed any almost symplectic manifold ) has a natural volume form.
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