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?

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.

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

Every soul has to follow the path, as explained by the Jinas and revived by the tirthankaras, to attain complete liberation or nirvana.

*( BCT1 ) Every complete metric space is a Baire space.

Every time Emperor Wen sent generals out on battles, he required them to follow the complete battle plans that he had drafted, and even the dates for battles needed approval from the emperor.

Every use of a bank machine, payment by credit card, use of a phone card, call from home, checked out library book, rented video, or otherwise complete recorded transaction generates an electronic record.

Every logical calculation, from the simplest to the most complex, must complete in one clock cycle.

Every time he is thrown out of the house, he is shown wearing the same shirt although he does not always wear it when he is thrown out ( the producers never shot a second sequence with Jazz being thrown out of the house, only adjusting the original scene for time purposes ; an exception is in the episode " Community Action ", where Jazz was thrown out along with a lifesize cardboard cut-out of Bill Cosby, complete with a blooper showing Jeff Townes reshooting his flying off the house several times ).

Every two years, before migration, the adult Common Crane undergoes a complete moult, remaining flightless for six weeks, until the new feathers grow.

*( BCT1 ) Every complete metric space is a Baire space.

Every complete musical piece projects a single key and ultimately a single stufe ( the tonic ).

Every student who intents to pass the matura is required to complete four exams:

Every projective variety is complete, but not vice versa.

" Every opening phase of a Canadian operation was a complete success and the staff works a mathematical masterpiece … the Canadian Army never followed up their opening successes to reach a complete victory.

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 5 years, each Commission conducts a complete review of all constituencies in its part of the United Kingdom.

Every month, a employer must file a complete IR348 Employer monthly schedule with the IRD, stating the income and deductions of each employee.

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

Every Skolem theory is model complete, i. e. every substructure of a model is an elementary substructure.

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 year on the third Saturday in September, German-Americans celebrate the Annual Steuben Parade on Fifth Avenue and an Oktoberfest-style beer fest complete with food and live music in Central Park.