** 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.

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

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

Every non-empty totally ordered set is directed.

Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere ( unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point ).

* 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.

# ( Accessibility Property ) Every non-empty feasible set X contains an element x such that X

Every non-empty file must have at least one fork, and depending on the file system, a file may have one or more other associated forks, which in turn may contain primary data integral to the file, or just metadata.

: Every set has a choice function.

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.

** Well-ordering theorem: Every set can be well-ordered.

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

** Antichain principle: Every partially ordered set has a maximal antichain.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

* 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 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 corporation, whether financial or union, as well as every division of the administration, were set up as branches of the party, the CEOs, Union leaders, and division directors being sworn-in as section presidents of the party.

Every DNS zone must be assigned a set of authoritative name servers that are installed in NS records in the parent zone, and should be installed ( to be authoritative records ) as self-referential NS records on the authoritative name servers.

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.

# " Personality " Argument: this argument is based on a quote from Hegel: " Every man has the right to turn his will upon a thing or make the thing an object of his will, that is to say, to set aside the mere thing and recreate it as his own ".

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 singleton set

Every processor or processor family has its own machine code instruction set.

Every set is a class, no matter which foundation is chosen.

* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

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 >.

Every chance I got I left the hotel to visit Lilly.

Every January 19, in the early hours of the morning, the person made a toast of cognac to Poe's original grave marker and left three roses.

If the book itself, Animal Farm, had left any doubt of the matter, Orwell dispelled it in his essay Why I Write: ' Every line of serious work that I ’ ve written since 1936 has been written directly or indirectly against Totalitarianism ... dot, dot, dot, dot.

Every pixel but the first is replaced by the difference to its left neighbor.

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 left ideal I in R is finitely generated, i. e. there exist elements a < sub > 1 </ sub >, ..., a < sub > n </ sub > in I such that I = Ra < sub > 1 </ sub > + ... + Ra < sub > n </ sub >.

Every one of the features encompassed by the theory still requires a reason for it to be maintained after hominids left the aquatic environment.

Every year since 1880, the organizations of the French left have held a demonstration in this symbolic place during the last week of May.

" Every part of his body has been replaced, with the exception of " all the diodes down left side ," which have been giving him severe pain for the whole of his existence.

Every week, the newest publication of Memín begins where the last publication had left off.

Every actor provided multiple voices, although the distinctive baritone of Earle Hyman ( Panthro ) left the actor providing only very occasional guest voices in comparison with his fellow performers.

Every factual error left in the published records is officially considered to be a failure by the responsible officer.

Every week, he left her food near the statue, which was placed outside the house, but nobody knew he was doing it.

Every one of them left a trace in the school's history.

Every seat in the ballpark is green, except for two-one in left field which marks the spot of Cal Ripken's 278th career homerun, breaking Ernie Banks'all time record among shortstops, and one in right field, which marks the spot of Eddie Murray's 500th career homerun.

Every chain either has the middle tile on the outside ( O ) or the inside ( I ) of the chain ; if it is arranged so that the indicator tile is to the right of the middle tile, then the position of the ring segment on the indicator tile can either be the upper left ( UL ), upper right ( UR ), lower left ( LL ) or lower right ( LR ) corner.

Every day, she saw trucks loaded with Jews heading to the railway station from where the trains left for concentration camps.

Every simple ring R with unity is both left and right primitive.

Every act of government was left to the individual States.

Every item has been left by a pilgrim who reports being healed at the basilica.

Every member of the Queen's Royal Hussars of the United Kingdom light cavalry wears the Maid of Warsaw, the crest of the City of Warsaw, on the left sleeve of his No. 2 ( Service ) Dress.

Every Samhain, the phantom Aillén mac Midgna, or Aillén the Burner, would terrorize Tara, playing music on his harp that left every warrior helpless.