** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

# Every open cover of A has a finite subcover.

* Countably compact: Every countable open cover has a finite subcover.

# Every finite and contingent being has a cause.

Every finite simple group is isomorphic to one of the following groups:

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

Every finite tree structure has a member that has no superior.

Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to.

* Every finite tree with n vertices, with, has at least two terminal vertices ( leaves ).

Every finite group of exponent n with m generators is a homomorphic image of B < sub > 0 </ sub >( m, n ).

Every known Sierpinski number k has a small covering set, a finite set of primes with at least one dividing k · 2 < sup > n </ sup >+ 1 for each n > 0.

Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.

: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index.

Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero.

Every finite group has a composition series, but not every infinite group has one.

* Every finite or cofinite subset of the natural numbers is computable.

* Every subset of may be covered by a finite set of positive orthants, whose apexes all belong to

* Every finite subextension of F / k is separable.

Every finite ordinal ( natural number ) is initial, but most infinite ordinals are not initial.

* Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.

* Every commutative semisimple ring must be a finite direct product of fields.

* Pseudocompact: Every real-valued continuous function on the space is bounded.

* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.

* Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.

* Every continuous function f: → R is bounded.

This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded.

For example, to study the theorem “ Every bounded sequence of real numbers has a supremum ” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.

Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.

Every three years the Company makes an award to the three buildings or structures, in the area bounded by the M25 motorway, which respectively embody the most outstanding example of brickwork, of slated or tiled roof and of hard-surface tiled wall and / or floor.

Every Polish town was bounded to put up a quantity of soldiers-this was a conspicuous sign of a power of a given town how much soldiers it had to put up.

Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element.

Every bounded linear transformation from a normed vector space to a complete, normed vector space can be uniquely extended to a bounded linear transformation from the completion of to.

Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.

Every subset of a totally bounded space is a totally bounded set ; but even if a space is not totally bounded, some of its subsets still will be.

* Every compact set is totally bounded, whenever the concept is defined.

* Every totally bounded metric space is bounded.

Every compact metric space is totally bounded.

* Every finite set of points is bounded

* Every relatively compact set in a topological vector space is bounded.

Every topological vector space X gives a bornology on X by defining a subset to be bounded iff for all open sets containing zero there exists a with.

Every ( bounded ) convex polytope is the image of a simplex, as every point is a convex combination of the ( finitely many ) vertices.