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 arithmetical subset of Cantor space of < sup >( or?

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.

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 building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space.

* Every Polish space is homeomorphic to a G < sub > δ </ sub > subspace of the Hilbert cube, and every G < sub > δ </ sub > subspace of the Hilbert cube is Polish.

Every Hilbert space has this property.

( 5 ) Every continuous affine isometric action of G on a real Hilbert space has a fixed point ( property ( FH )).

Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L < sup >∞</ sup >( X ) for some standard measure space ( X, μ ) and conversely, for every standard measure space X, L < sup >∞</ sup >( X ) is a von Neumann algebra.

Every correspondence prescription between phase space and Hilbert space, however, induces its own proper-product.

* Every smooth Hilbert manifold can be smoothly embedded onto an open subset of the model Hilbert space.

* Every homotopy equivalence between two Hilbert manifolds is homotopic to a diffeomorphism.

Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.

Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics.

Every human mind inherits a complement of species-specific “ elementary ideas ” ( Elementargedanken ), and hence the minds of all people, regardless of their race or culture, operate in the same way.

Every path from back door to barn was covered by a grape-arbor, and every yard had its fruit trees.

Every morning early, in the summer, we searched the trunks of the trees as high as we could reach for the locust shells, carefully detached their hooked claws from the bark where they hung, and stabled them, a weird faery herd, in an angle between the high roots of the tulip tree, where no grass grew in the dense shade.

Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.

Every pitch in the game brought forth a howl from the enraptured audience and every fly ball the visitors dropped ( and because their right fielder was still a little fuzzy from drink, they dropped many ) called forth yelps of derision.

Every line of written text is a mere reflection of references from any of a multitude of traditions, or, as Barthes puts it, " the text is a tissue of quotations drawn from the innumerable centres of culture "; it is never original.

Every book followed a different train of thought, from inspiration to solution, according to her autobiography.

Every time a speech sound is produced for a given phoneme, it will be slightly different from other utterances, even for the same speaker.

: Every non-empty set A contains an element B which is disjoint from A.

Every year both clubs play the " Klassieker " (" The Classic "), a derby match between the teams from the two largest cities of the Netherlands.

Four other hits from the album made the Pop top 20: " Every Heartbeat " ( No. 2 ), " That's What Love Is For " ( No. 7 ), " Good For Me " ( No. 8 ), and " I Will Remember You " ( No. 20 ).

* Revised Code of Washington 9. 12. 010: " Every person who brings on his or her own behalf, or instigates, incites, or encourages another to bring, any false suit at law or in equity in any court of this state, with intent thereby to distress or harass a defendant in the suit, or who serves or sends any paper or document purporting to be or resembling a judicial process, that is not in fact a judicial process, is guilty of a misdemeanor ; and in case the person offending is an attorney, he or she may, in addition thereto be disbarred from practicing law within this state.

Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed.

Every year, the European Joint Conferences on Theory and Practice of Software ( ETAPS ) sponsors the International Conference on Compiler Construction ( CC ), with papers from both the academic and industrial sectors.

Every Halloween, Dabney House conducts the infamous " Millikan pumpkin-drop experiment " from the top of Millikan Library, the highest point on campus.

* 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.

Every year many teams from all over Asia converge in Tokyo to compete.

Every argues that " the disparagement of myth in our own civilization " stems partly from objections to perceived idolatry, objections which intensified in the Reformation, both among Protestants and among Catholics reacting against the classical mythology revived during the Renaissance.

Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed.

In Norse mythology, Draupnir ( Old Norse " the dripper ") is a gold ring possessed by the god Odin with the ability to multiply itself: Every ninth night eight new rings ' drip ' from Draupnir, each one of the same size and weight as the original.

Every time a diode switches from on to off or vice versa, the configuration of the linear network changes.

* Every June, Ellensburg hosts Dachshunds on Parade, an event that draws Dachshund dog owners from all over the Northwest.

For example the words of " I'm a Man You Don't Meet Every Day " ( Roud 975 ) are known from a broadside in the Bodleian Library.

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.