** 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 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 finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation.

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.

* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.

* Every topological group is completely regular.

Every group can be trivially made into a topological group by considering it with the discrete topology ; such groups are called discrete groups.

Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.

Every subgroup of a topological group is itself a topological group when given the subspace topology.

Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.

Every local field is isomorphic ( as a topological field ) to one of the following:

* Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval 1.

Every directed acyclic graph has a topological ordering, an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge.

Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological group.

Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.

* Every constant function between topological spaces is continuous.

Every topological group is an H-space ; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.

Every interior algebra can be represented as a topological field of sets with its interior and closure operators corresponding to those of the topological space.

Every separable topological space is ccc.

Every metric space which is ccc is also separable, but in general a ccc topological space need not be separable.

Every locally compact group which is second-countable is metrizable as a topological group ( i. e. can be given a left-invariant metric compatible with the topology ) and complete.