Page "Functor" ¶ 34
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## Some Related Sentences

Every and continuous
* 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 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.
* Pseudocompact: Every real-valued continuous function on the space is bounded.
Every contraction mapping is Lipschitz continuous and hence uniformly continuous ( for a Lipschitz continuous function, the constant k is no longer necessarily less than 1 ).
Every space filling curve hits some points multiple times, and does not have a continuous inverse.
* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions R, with the supremum norm.
Every uniformly continuous function between metric spaces is continuous.
Every continuous function on a compact set is uniformly continuous.
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 continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
* Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous.
Every embedding is injective and continuous.
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 compact Hausdorff space of weight at most ( see Aleph number ) is the continuous image of ( this does not need the continuum hypothesis, but is less interesting in its absence ).
Every place south of the Antarctic Circle experiences a period of twenty-four hours ' continuous daylight at least once per year, and a period of twenty-four hours ' continuous night time at least once per year.

Every and map
Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.
Every smooth ( or differentiable ) map φ: M N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:
Every distinct map projection distorts in a distinct way.
Every inner automorphism is indeed an automorphism of the group G, i. e. it is a bijective map from G to G and it is a homomorphism ; meaning ( xy )< sup > a </ sup >
* Every constant map is a plot.
Every regular map of varieties is continuous in the Zariski topology.
Every real m-by-n matrix yields a linear map from R < sup > n </ sup > to R < sup > m </ sup >.
Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.
Every local homeomorphism is a continuous and open map.
Every covering map is a semicovering, but semicoverings satisfy the " 2 out of 3 " rule: given a composition of maps of spaces, if two of the maps are semicoverings, then so also is the third.
* Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.
Every Möbius transformation is a bijective conformal map of the Riemann sphere to itself.
* The embedding theorem for Stein manifolds states the following: Every Stein manifold of complex dimension can be embedded into by a biholomorphic proper map.
Every issue of our periodical will therefore include one or more map supplements, and their design will guarantee a continuous and easily accessible supplement in easy-to-manage form with special regard for those who own Stielers Hand-Atlas, Berghaus ’ s Physical Atlas, and other map publications of the ( Perthes ) Institute.
Every map consists of numerous textured polygons carefully positioned in relation to one another.
Every summer, the town prepares for the one-week summer festival, " Finnsnes i Fest ", aiming to put Finnsnes on the map.
* Every map can be replaced by a cofibration via the mapping cylinder construction
* Every local diffeomorphism is also a local homeomorphism and therefore an open map.
Every element x of G gives rise to a tensor-preserving self-conjugate natural transformation via multiplication by x on each representation, and hence one has a map.
Every map has these two bases, but each map has a different pattern of fixed terrain features.

Every and f
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 isogeny f: A B is automatically a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.
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 )).

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