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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.
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Any and discrete
* Any discrete valuation ring.
Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.
Any discrete variable is a segmentation.
Any discrete emotion can be used in a persuasive appeal ; this may include jealousy, disgust, indignation, fear, blue, disturbed, haunted, and anger.
* Any finite abelian group, with the discrete topology.
Any group can be given the discrete topology.
Any Fuchsian group ( a discrete subgroup of SL ( 2, R )) is a Kleinian group, and conversely any Kleinian group preserving the real line ( in its action on the Riemann sphere ) is a Fuchsian group.
* Any countable discrete group with property ( T ) is finitely generated.
* Any discrete group is locally compact.
* Any discrete space is a 0-dimensional manifold.
Any and subset
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 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 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 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 subset of a meagre set is meagre ; any superset of a comeagre set is comeagre.
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 and Euclidean
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 square matrix with unit Euclidean norm is the average of two unitary matrices.
* Any reflection, translation and rotation is a global isometry on Euclidean spaces.
Any non-zero vector v divides the Euclidean space into two half-spaces bounding the hyperplane v < sup >∧</ sup > orthogonal to v, namely v < sup >+</ sup > and v < sup >−</ sup >.
Any geometry found inside the square representing the Manhattan distance from the corner undergoes an additional check to see if the same geometry lies outside the quarter-circle radius representing the Euclidean distance.
Any finite set of lines in the Euclidean plane has a combinatorially equivalent arrangement in the hyperbolic plane ( e. g. by enclosing the vertices of the arrangement by a large circle and interpreting the interior of the circle as a Klein model of the hyperbolic plane ).
* Any Euclidean space is contractible, as is any star domain on a Euclidean space.
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 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 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.
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