Page "Topological group" ¶ 19
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Every and topological
* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.
* 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 topological group is completely regular.
Every group can be trivially made into a topological group by considering it with the discrete topology ; such groups are called discrete groups.
Every subgroup of a topological group is itself a topological group when given the subspace topology.
Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.
Every local field is isomorphic ( as a topological field ) to one of the following:
* 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.
Every directed acyclic graph has a topological ordering, an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge.
Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological 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 constant function between topological spaces is continuous.
Every topological group is an H-space ; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.
Every interior algebra can be represented as a topological field of sets with its interior and closure operators corresponding to those of the topological space.
Every separable topological space is ccc.
Every metric space which is ccc is also separable, but in general a ccc topological space need not be separable.
Every locally compact group which is second-countable is metrizable as a topological group ( i. e. can be given a left-invariant metric compatible with the topology ) and complete.

Every and group
Every finite simple group is isomorphic to one of the following groups:
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 group G acts on G, i. e. in two natural but essentially different ways:, or.
Every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters.
* Every Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )
Every instrumental group ( or section ) has a principal who is generally responsible for leading the group and playing orchestral solos.
* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.
While he admits the existence of caste-based discrimination, he writes that " Every social group cannot be regarded as a race simply because we want to protect it against prejudice and discrimination ".
Every local group is required to have a seneschal who reports to the kingdom's seneschal.
Every local group is required to have one.
Every summer the group gathers in Newport, RI for week long dance training, seaside teas, and evenings enjoying the splendors of the Gilded Age.
* 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 synset contains a group of synonymous words or collocations ( a collocation is a sequence of words that go together to form a specific meaning, such as " car pool "); different senses of a word are in different synsets.
:" Every group is naturally isomorphic to its opposite group "
Every ten years, when the general census of population takes place, each citizen has to declare which linguistic group they belong or want to be aggregated to.

Every and can
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.
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 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 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 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 >.

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