* Every recursively enumerable set is arithmetical.

Every arithmetical set is implicitly arithmetical ; if X is arithmetically defined by φ ( n ) then it is implicitly defined by the formula

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Every infinite game in which is a Borel subset of Baire space is determined.

# Every infinite subset of X has a complete accumulation point.

# Every infinite subset of A has at least one limit point in A.

* Limit point compact: Every infinite subset has an accumulation point.

Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.

* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.

* Every separable metric space is homeomorphic to a subset of the Hilbert cube.

* 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 separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

* Every separable metric space is isometric to a subset of the

Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.

* Every subset of Baire space or Cantor space is an open set in the usual topology on the space.

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable ( and therefore T4 ) and second countable.

It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.

* Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

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

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

** Every vector space has a basis.

** Every Tychonoff space has a Stone – Čech compactification.

* Theorem Every reflexive normed space is a Banach space.

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

* 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 compact metric space is separable.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

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

Every compact metric space is complete, though complete spaces need not be compact.

Every point in three-dimensional Euclidean space is determined by three coordinates.

Every node on the Freenet network contributes storage space to hold files, and bandwidth that it uses to route requests from its peers.

Every space filling curve hits some points multiple times, and does not have a continuous inverse.

* Every Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )

Every vector space has a basis, and all bases of a vector space have the same number of elements, called the dimension of the vector space.

Every normed vector space V sits as a dense subspace inside a Banach space ; this Banach space is essentially uniquely defined by V and is called the completion of V.

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

Every bijective function g has an inverse g < sup >− 1 </ sup >, such that gg < sup >− 1 </ sup > = I ;

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every atom across this plane has an individual set of emission cones .</ p > < p > Drawing the billions of overlapping cones is impossible, so this is a simplified diagram showing the extents of all the emission cones combined.

Every ketose will have 2 < sup >( n-3 )</ sup > stereoisomers where n > 2 is the number of carbons.

Every aldose will have 2 < sup >( n-2 )</ sup > stereoisomers where n > 2 is the number of carbons.

Every time an MTA receives an email message, it adds a < tt > Received </ tt > trace header field to the top of the header of the message, thereby building a sequential record of MTAs handling the message.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

< li > Every positive definite matrix is invertible and its inverse is also positive definite.

Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n, and with the exception of < var > n </ var > = 1, < var > n </ var > must end in 0, 2, 3, 5, 7, or 8.

Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.

* Every linear combination of its components Y = a < sub > 1 </ sub > X < sub > 1 </ sub > + … + a < sub > k </ sub > X < sub > k </ sub > is normally distributed.

Every sedenion is a real linear combination of the unit sedenions 1, < var > e </ var >< sub > 1 </ sub >, < var > e </ var >< sub > 2 </ sub >, < var > e </ var >< sub > 3 </ sub >, ..., and < var > e </ var >< sub > 15 </ sub >,

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.