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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.

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## Some Related Sentences

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__

**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__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__

__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__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

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__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__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__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

**.**

__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__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__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__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__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__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__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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