* Any finite topological space, including the empty set, is compact.

# Any collection of closed subsets of X with the finite intersection property has nonempty intersection.

Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.

* Any two finite fields with the same number of elements are isomorphic.

# Any finite straight line can be extended in a straight line.

* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

Any meaningful theory of quantum gravity that makes sense and is predictive at all energy scales must have some deep principle that reduces the infinitely many unknown parameters to a finite number that can then be measured.

Any topological space which is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself.

Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.

Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.

Any digital modulation scheme uses a finite number of distinct signals to represent digital data.

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.

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.

Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions.

* Any finite contiguous subsequence of a word

Any finite extension of a finite field is a cyclic extension.

Any structure implicit in the finite entropy against a quantum description could then be totally washed out by the huge injection of uncertainty this projection represents.

Any Calabi – Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi – Yau manifold.

Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions.

Any Fredholm operator has an index, defined as the difference between the ( finite ) dimension of the kernel of D ( solutions of Df = 0 ), and the ( finite ) dimension of the cokernel of D ( the constraints on the right-hand-side of an inhomogeneous equation like Df

* Any category with finite products is monoidal with the product as the monoidal product and the terminal object as the unit.

Any company incorporating Bluetooth wireless technology into products, using the technology to offer goods and services or simply re-branding a product with Bluetooth technology must become a member of the Bluetooth SIG.

Any heat that isn't translated into work is normally considered a waste product and is removed from the engine either by an air or liquid cooling system.

* Any Cartesian product of intervals b and d is Lebesgue measurable, and its Lebesgue measure is ( b − a )( d − c ), the area of the corresponding rectangle.

Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction.

Any polynomial may be decomposed into the product of a constant by a product of irreducible polynomials.

Any ' defect ' in the ' product ' that is not reported is fraud.

Any age statement on a bottle of Scotch whisky, expressed in numerical form, must reflect the age of the youngest whisky used to produce that product.

Any alcoholic product, when served as a chilled shot, may be more palatable to the consumer.

* Any petroleum product or fat ( including those used in cooking ) that is a soft solid at room temperature

Any product line associated with the show had to be educational, inexpensive, and could not be advertised during the show's airings.

Any positive divisor of is a product of prime divisors of raised to some power.

Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:

Any n × n permutation matrix can be constructed as a product of no more than n − 1 transpositions.

Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections.

Any rotation matrix of size n × n can be constructed as a product of at most n ( n − 1 )/ 2 such rotations.

* Any other aspect of the product or service that is subject to regulatory compliance

Any given slave will only respond to particular messages, as defined by its product documentation.

Any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues.

* Any two objects X and Y of C have a product X × Y in C.

: Any type of product or service — pharmaceuticals, building construction, vehicles, electronics, computer software, financial services, etc.

Any abelian variety is isogenous to a product of simple abelian varieties.

Any efficiency learned from one product can be applied to the other products.