* 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 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 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 Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous.

Every distinct map projection distorts in a distinct way.

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 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 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 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.

Every information exchange between living organisms — i. e. transmission of signals that involve a living sender and receiver can be considered a form of communication ; and even primitive creatures such as corals are competent to communicate.

Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form.

* Every regular language is context-free because it can be described by a context-free grammar.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form.

Every real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every entire function can be represented as a power series that converges uniformly on compact sets.

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.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

Every sequence can, thus, be read in three reading frames, each of which will produce a different amino acid sequence ( in the given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ).

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every species can be given a unique ( and, one hopes, stable ) name, as compared with common names that are often neither unique nor consistent from place to place and language to language.

Every morpheme can be classified as either free or bound.

Every use of modus tollens can be converted to a use of modus ponens and one use of transposition to the premise which is a material implication.

Every document window is an object with which the user can work.

Every adult, healthy, sane Muslim who has the financial and physical capacity to travel to Mecca and can make arrangements for the care of his / her dependants during the trip, must perform the Hajj once in a lifetime.

Every ordered field can be embedded into the surreal numbers.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

* 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 >.