Every time I closed my eyes, I saw Gray Eyes rushing at me with a knife.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.

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

* 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 closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.

Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G

* Every closed nowhere dense set is the boundary of an open set.

Every map that is injective, continuous and either open or closed is an embedding ; however there are also embeddings which are neither open nor closed.

Every base he closed resulted in a new construction project elsewhere to expand another base, relocation of forces projects and other related spending.

Every year the central business district ( with corners at the Municipal Building, Grand Street Fire House and Croton-Harmon High School ) is closed to automobile traffic for music, American food, local fund raisers, traveling, and local artists.

The generalized Poincaré conjecture states that Every simply connected, closed n-manifold is homeomorphic to the n-sphere.

Every closed subspace of a reflexive space is reflexive.

Every closed 3-manifold has a prime decomposition: this means it is the connected sum of prime three-manifolds ( this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds ).

: 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 integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.

* 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 October the high street is closed for the two Saturdays either side of 11 October for the Marlborough Mop Fair.

Every closed point of Hilb ( X ) corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.

Every homeomorphism is open, closed, and continuous.

Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is,

* Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem

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

Every subgroup of a free abelian group is itself free abelian, which is important for the description of a general abelian group as a cokernel of a homomorphism between free abelian groups.

Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

* Every subgroup and quotient group of a locally cyclic group is locally cyclic.

* Every triangle group T is a discrete subgroup of the isometry group of the sphere ( when T is finite ), the Euclidean plane ( when T has a Z + Z subgroup of finite index ), or the hyperbolic plane.

Every compact Lie group G of dimension > 0 has a subgroup isomorphic to the circle group.

Recall that a subsemigroup G of a semigroup S is a subgroup of S ( also called sometimes a group in S ) if there exists an idempotent e such that G is a group with identity element e. A semigroup S is group-bound if some power of each element of S lies in some subgroup of S. Every finite semigroup is group-bound, but a group-bound semigroup might be infinite.

Every subgroup is organized around a set of brothers, each of whom is often married to a group of sisters.

Every finite group is a subgroup of the mapping class group of a closed, orientable surface, moreover one can realize any finite group as the group of isometries of some compact Riemann surface.

Every proper subgroup of G can be assumed a solvable group, meaning that much theory of such subgroups could be applied.

Every closed subgroup of a locally compact group is locally compact.

Every quasinormal subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups.

Every quasinormal subgroup is conjugate permutable.

Every normal subgroup is quasinormal, because, in fact, a normal subgroup commutes

Every group has itself ( the improper subgroup ) and the trivial subgroup as two of its fully characteristic subgroups.

* Every quotient of an amenable goup is amenable.

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 detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every taxpayer is well aware of the vast size of our annual defense budget and most of our readers also realize that a large portion of these expenditures go for military electronics.

Every single problem touched on thus far is related to good marketing planning.

Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.

Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available.

Every man in every one of these houses is a Night Rider.

Every library borrower, or at least those whose taste goes beyond the five-cent fiction rentals, knows what it is to hear the librarian say apologetically, `` I'm sorry, but we don't have that book.

Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.

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 natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).

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 root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every year, on the last Sunday in April, there is an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.