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 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 subset of B is also well-ordered.

:* Any subset of the identity relation on X has equivalence classes that are the singletons of X.

Any one language has only a subset of the aspectual distinctions attested in the world's languages, and some languages ( such as Standard German ; see below ) do not have aspects.

Any measurable subset of a null set is itself a null set.

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 subset of resource, but typically at least the processor state, may be associated with each of the process ' threads in operating systems that support threads or ' daughter ' processes.

Any function can be restricted to a subset of its domain.

* Any subset of a meagre set is meagre ; any superset of a comeagre set is comeagre.

Any discrete subset of Euclidean space is countable, since the isolation of each of its points ( together with the fact the rationals are dense in the reals ) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many.

Any proposed inverse R of F ( reconstruction formula, in the lingo ) would have to map to some subset of.

Any action of the group by continuous affine transformations on a compact convex subset of a ( separable ) locally convex topological vector space has a fixed point.

Any induced subgraph of a comparability graph is itself a comparability graph, formed from the restriction of the partial order to a subset of its elements.

* Any directed set A may be made into a Cauchy space by declaring a filter F to be Cauchy if, given any element n of A, there is an element U of F such that U is either a singleton or a subset of the tail

Any subset of a group closed under multiplication, under inversion and under identity ( i. e.

Any given neuron only responds to a subset of stimuli within its receptive field.

* Any open subset of an n-manifold is a n-manifold with the subspace topology.

Any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral.

* Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.

Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan.

# Any finite-dimensional unital associative algebra A has a natural homomorphism to its own endomorphism ring End ( A ).

* Any finite-dimensional vector space is nuclear, because any operator on a finite-dimensional vector space is nuclear.