: Given any family of nonempty sets, their Cartesian product is a nonempty set.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given that John of Worcester wrote his chronicle after the eruption of the Canterbury – York supremacy struggle, the story of Ealdred renouncing any claims to Worcester needs to be considered suspect.

Given the absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

We have seen a few cavities of the appropriate size and shape for ivory-bills, but these can be old, or exceptionally large Pileated Woodpecker cavities, or mammal-enlarged Pileated Woodpecker cavities .… Given the results, it is unlikely a population of any meaningful size of Ivory-billed Woodpeckers exists in south Florida.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Given the above-mentioned problems, regulators face the challenging task of regulating a market that is changing very rapidly, without stifling any type of innovation, and without improperly disadvantaging any competitor.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

: Given any positive number ε, there is a sequence

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).

Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.

Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.

Given any topological space X, the zero sets form the base for the closed sets of some topology on X.

Given a locally compact topological field K, an absolute value can be defined as follows.

Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.

Given a complex vector bundle V over a topological space X,

Given a point x of a topological space X, and two maps f, g: X → Y ( where Y is any set ), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal ;

Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module.

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.

Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain ( also called fundamental region ) for this action is a set D of representatives for the orbits.

Given a compact topological space X, the topological K-theory K < sup > top </ sup >( X ) of ( real ) vector bundles over X coincides with K < sub > 0 </ sub > of the ring of continuous real-valued functions on X.

Given a topological group G, the Bohr compactification of G is a compact Hausdorff topological group Bohr ( G ) and a continuous homomorphism

Given a topological space and a subset of, the subspace topology on is defined by

Given a topological space X = 〈 X, T 〉 one can form the power set Boolean algebra of X:

Given a field of sets the complexes form a base for a topology, we denote the corresponding topological space by.

Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interior and closure.

Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra ( which form a base for a topology ).