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Page "Path (topology)" ¶ 25
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Any and topological
* Any finite topological space, including the empty set, is compact.
Any topological space which is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself.
* Any topological space which is the union of a countable number of separable subspaces is separable.
* 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 ordinal number can be made into a topological space by endowing it with the order topology ( since, being well-ordered, an ordinal is in particular totally ordered ): in the absence of indication to the contrary, it is always that order topology which is meant when an ordinal is thought of as a topological space.
* Any continuous map of topological spaces determines a sheaf of sets.
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 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 DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time.
* Any uniform space ( hence any metric space, topological vector space, or topological group ) is a Cauchy space ; see Cauchy filter for definitions.
Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space.
Any topological space which deformation retracts to a point is contractible.
1 ) Any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold with no differentiable structure ( so cannot be smoothed ).
Any point on the boundary between these two sets is both a left and right topological divisor of zero.

Any and space
* 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 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 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 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 X
# Any collection of closed subsets of X with the finite intersection property has nonempty intersection.
:* Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y ;
:* Any subset of the identity relation on X has equivalence classes that are the singletons of X.
Any attempt to define it ( X is good if it has property Y ) will simply shift the problem ( Why is Y-ness good in the first place?
Any other responses will allow X to force the win.
* quantification Any morphism f: X → Y in a category with pullbacks induces a monotonous map acting by pullbacks ( A monotonous map is a functor if we consider the preorders as categories ).
Any collection of subsets of a set X satisfying these properties forms a base for the closed sets of a topology on X.
Any formal series with has a composition inverse provided is an invertible element of R. The coefficients are found recursively from the above formula for the coefficients of a composition, equating them with those of the composition identity X ( that is 1 at degree 1 and 0 at every degree greater than 1 ).
* Any two objects X and Y of C have a product X × Y in C.
:: Theorem ( Lefschetz theorem on ( 1, 1 )- classes ) Any element of H < sup > 2 </ sup >( X, Z ) ∩ H < sup > 1, 1 </ sup >( X ) is the cohomology class of a divisor on X.
:* Any two simplices in X are contained in a common apartment.
Any simplicial automorphism φ of X defines a permutation π of Z / n Z such that label ( φ ( M ))
Any application that supports the Secure Socket Layer ( SSL ) can make use of certificates signed by CAcert, as can any application that uses X. 509 certificates, e. g. for encryption or code signing and document signatures.
Any term p < sub > k </ sub > X < sup > k </ sup > with zero coefficient, p < sub > k </ sub > = 0, may be omitted.
Any ξ ∈ gives rise to a canonical vertical vector field X < sub > ξ </ sub > by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ.
Wiley published several books during the course of his career, including Serenity, A Boxing Memoir ; Why Black People Tend To Shout ; and By Any Means Necessary: The Trials and Tribulations of Making Malcolm X, with Spike Lee.

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