Any subset of R < sup > n </ sup > ( with its subspace topology ) that is homeomorphic to another open subset of R < sup > n </ sup > is itself open.

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 decision in allocating capital is likewise: there is an opportunity cost of capital, or a hurdle rate, defined as the expected rate one could get by investing in similar projects on the open market.

Any legal wrestler is open to attack from any direction at any time, including when they are downed, as long as they are within the ring area enclosed by the ring ropes.

Any canal built open to all nations on equal terms.

Eric Raymond extends this principle in support of open source security software, saying, " Any security software design that doesn't assume the enemy possesses the source code is already untrustworthy ; therefore, never trust closed source.

* 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 finite subcover of this cover must be bounded, because all balls in the subcover are contained in the largest open ball within that subcover.

Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane.

Any relatively flat, open space can be used as a terrain.

* Aranym Atari Running on Any Machine: an open source emulator / virtual machine that can run Atari GEM applications

According to the historian of science Norwood Russell Hanson: There is no bilaterally-symmetrical, nor excentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent. Any path — periodic or not, closed or open — can be represented with an infinite number of epicycles.

( 2 ) Any non-profit corporation, civic association, or governmental entity which has buildings and grounds open to the public may request for the appointment of constables to serve as law enforcement officers in order to protect life and property.

* Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.

Any ordinal is, of course, an open subset of any further ordinal.

* Any open set is trivially a G < sub > δ </ sub > set

Any competitors desiring their names to be attached to their contributions will please give permission to the Committee to open the envelopes inscribed with their mottoes.

Any open cover of a paracompact space is numerable.

Any future attempt to recover his throne would have to be with the open support of the English king.

Any point x gives rise to a continuous function p < sub > x </ sub > from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p < sub > x </ sub >( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.

Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs.

# 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 real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian.

* 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 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.

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 “ 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.