Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).

Every homomorphism f: G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and.

Every locally constant function from the real numbers R to R is constant by the connectedness of R. But the function f from the rationals Q to R, defined by f ( x ) = 0 for x < π, and f ( x ) = 1 for x > π, is locally constant ( here we use the fact that π is irrational and that therefore the two sets

* Every continuous function f: → R is bounded.

Every empty function is constant, vacuously, since there are no x and y in A for which f ( x ) and f ( y ) are different when A is the empty set.

Theorem Every self-adjoint f in A * can be written as f

Every order-preserving self-map f of a cpo ( P, ⊥) has a least fixpoint.

Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.

Every morphism f: G → H in Grp has a category-theoretic kernel ( given by the ordinary kernel of algebra ker f =

Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary.

Every deterministic complexity class ( DSPACE ( f ( n )), DTIME ( f ( n )) for all f ( n )) is closed under complement, because one can simply add a last step to the algorithm which reverses the answer.

* Every point x of X is isolated in its fiber f < sup >− 1 </ sup >( f ( x )).

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

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

* Every representable functor C → Set preserves limits ( but not necessarily colimits ).

Every universal cover p: D → X is regular, with deck transformation group being isomorphic to the fundamental group.

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 functor F: D → E induces a functor F < sup > C </ sup >: D < sup > C </ sup > → E < sup > C </ sup > ( by composition with F ).

# Every subset of an independent set is independent, i. e., for each E ' ⊆ E, E ∈ I → E ' ∈ I.

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

Every year, it invites entries of unpublished African-inspired stories written for an audience of 8 to 11-year-olds ( Category A ) or 12 to 15-year-olds ( Category B ) and by writers aged 18 or below ( the Rising Writer Prize ).

Every year since 1982, the W. C. Handy Music Festival is held in the Florence / Sheffield / Muscle Shoals area, featuring blues, jazz, country, gospel, rock music and R & B.

Every finite group of exponent n with m generators is a homomorphic image of B < sub > 0 </ sub >( m, n ).

The Pips recorded a second version of " Every Beat " with Bobby Robinson as the producer, and the song became a # 1 R & B and # 6 pop hit.

Every single that Bell produced for the Stylistics was a Top Ten R & B hit, and several —" You Are Everything ", " Betcha by Golly Wow!

" " Every Day " hit number two on the R & B charts, and sparked another LP — 1957 The Greatest!

Every other fret remains unaffected ( e. g. the 7th fret of an E-string will still play a B note for any capo position at or below the 7th fret ), and thus a performer does not need to adjust for or relearn the entire fretboard as they might with retuning.

B ' nai B ' rith also publishes program guides for local Jewish education programs and each year sponsors " Unto Every Person There is a Name ".

This means that A is not necessarily better than B, and B is not necessarily better than C. Every class is independent of the others.

Every degree of freedom in the energy is quadratic and, thus, should contribute k < sub > B </ sub > T to the total average energy, and k < sub > B </ sub > to the heat capacity.

* Every Borel set B ⊆ R < sup > n </ sup > is μ-measurable in the sense of Carathéodory's criterion: for every A ⊆ R < sup > n </ sup >,

* Every audience member had his or her own A / B / C / D keypad.

She appeared in Plaza Suite, Every Little Crook and Nanny, B. S.

* Stockham, Alice B Tokology: A Book for Every Woman, Sanitary Publishing Co., Illinois, 1886, with eight Color Plates in a sleeve inside the back cover.

Every other day, starting on the third day, the player can go after an A. P. B.

Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.

B. Lippincott Company, but was reprinted once in 1959 by Pocket Books under the title Mr. Adam Was Wanted By Every Woman in the World.

Every weekend, the band would play the opening set for touring bands such as Earth, Wind and Fire, Stevie Wonder, James Brown and Kool & The Gang, and the charangueros became huge fans of both the music and stage shows of North American R & B.

Every injector they tested had detectable blood in a quantity sufficient to pass on a virus such as hepatitis B.

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.