[permalink] [id link]

Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.

from
Wikipedia

## 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__

__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__subgroup of

**a**

__topological__

**group**

**is**itself

**a**

__topological__

**group**when given the subspace topology

**.**

*

__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 ring

__Every__Boolean algebra

**(**A, ∧, ∨) gives rise

**to**

**a**

__ring__

**(**A, +, ·) by defining

**a**+ b :=

**(**

**a**∧ ¬ b

**)**∨

**(**b ∧ ¬

**a**

**)**=

**(**

**a**∨ b

**)**∧ ¬(

**a**∧ b

**)**

**(**this operation

**is**called symmetric difference

**in**the case of sets

**and**XOR

**in**the case of logic

**)**

**and**

**a**· b :=

**a**∧ b

**.**The zero element of this

__ring__coincides

**with**the 0 of the Boolean algebra ; the multiplicative identity element of the

__ring__

**is**the 1 of the Boolean algebra

**.**

In Norse mythology, Draupnir

**(**Old Norse " the dripper ")**is****a**gold__ring__possessed by the god Odin**with**the ability**to**multiply itself:__Every__ninth night eight new rings ' drip ' from Draupnir, each one of the same size**and**weight as the original**.**__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__objective has

**a**different size

__ring__, so for every objective another condenser setting has

**to**be chosen

**.**

__Every__match must be assigned

**a**rule keeper known as

**a**referee, who

**is**the final arbitrator

**(**In multi-man lucha libre matches, two referees are used, one inside the

__ring__

**and**one outside ).

* In any

__ring__R,**a**maximal ideal**is**an ideal M that**is**maximal**in**the set of all proper ideals of R, i**.**e**.**M**is**contained**in**exactly 2 ideals of R, namely M itself**and**the entire__ring__R**.**__Every__maximal ideal**is****in**fact prime**.**__Every__prime ideal P

**in**

**a**Boolean

__ring__R

**is**maximal: the quotient

__ring__R / P

**is**an integral domain

**and**also

**a**Boolean

__ring__, so it

**is**isomorphic

**to**the field F < sub > 2 </ sub >, which shows the maximality of P

**.**Since maximal ideals are always prime, prime ideals

**and**maximal ideals coincide

**in**Boolean rings

**.**

__Every__polynomial

**in**can be factorized into polynomials that are irreducible over F

**.**This factorization

**is**unique up

**to**permutation of the factors

**and**the multiplication of the factors by nonzero constants from F

**(**because the

__ring__of polynomials over

**a**field

**is**

**a**unique factorization domain whose units are the nonzero constant polynomials ).

Every and is

__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__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__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__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__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

**.**

0.198 seconds.