[permalink] [id link]

* 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.

from
Wikipedia

## Some Related Sentences

Every and finite

** Tukey's lemma:

__Every__non-empty collection**of**__finite__character has**a**maximal element with respect**to**inclusion**.**
Hilbert's example: "

**the**assertion that either there are**only**finitely many prime numbers or there are infinitely many " ( quoted**in**Davis 2000: 97 );**and**Brouwer's: "__Every__mathematical species is either__finite__or infinite**.**__Every__rational number / has two closely related expressions

**as**

**a**

__finite__continued fraction

**,**whose coefficients

**can**

**be**determined by applying

**the**Euclidean algorithm

**to**

**.**

__Every__

__finite__group

**of**exponent n with m generators is

**a**homomorphic image

**of**B < sub > 0 </ sub >( m

**,**n ).

__Every__known Sierpinski number k has

**a**small covering set

**,**

**a**

__finite__set

**of**primes with at least one dividing k · 2 < sup > n </ sup >+ 1 for each n > 0

**.**

__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__oriented prime closed 3-manifold**can****be**cut along tori**,**so that**the**interior**of**each**of****the**resulting manifolds has**a**geometric structure with__finite__volume**.**__Every__

__finite__or bounded interval

**of**

**the**real numbers that contains an infinite number

**of**

**points**must have at least one point

**of**accumulation

**.**

__Every__field

**of**either type

**can**

**be**realized

**as**

**the**field

**of**fractions

**of**

**a**Dedekind domain

**in**

**which**

**every**non-zero ideal is

**of**

__finite__index

**.**

__Every__process involving charged particles emits infinitely many coherent photons

**of**infinite wavelength

**,**

**and**

**the**amplitude for emitting any

__finite__number

**of**photons is zero

**.**

*****

__Every__finite-dimensional central simple algebra over

**a**

__finite__field must

**be**

**a**matrix ring over that field

**.**

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__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__

__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__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

**.**

0.215 seconds.