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* 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.

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## Some Related Sentences

Every and topological

*****

__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__

__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 space

__Every__Hilbert

__space__

**X**

**is**

**a**Banach

__space__because

**,**

**by**definition

**,**

**a**Hilbert

__space__

**is**complete with respect to

**the**norm associated with its inner product

**,**where

**a**norm and an inner product are said to be associated if for all x ∈

**X**

**.**

*****

__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__subset A

**of**

**the**vector

__space__

**is**contained within

**a**smallest convex set ( called

**the**convex hull

**of**A ), namely

**the**intersection

**of**all convex sets containing A

**.**

__Every__node on

**the**Freenet network contributes storage

__space__to hold files

**,**and bandwidth that it uses to route requests from its peers

**.**

*****

__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__vector

__space__has

**a**basis

**,**and all bases

**of**

**a**vector

__space__have

**the**same number

**of**elements

**,**called

**the**dimension

**of**

**the**vector

__space__

**.**

__Every__normed vector

__space__V sits as

**a**

**dense**

**subspace**inside

**a**Banach

__space__; this Banach

__space__

**is**essentially uniquely defined

**by**V and

**is**called

**the**completion

**of**V

**.**

Every and X

__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__linear combination

**of**its components Y =

**a**< sub > 1 </ sub >

__X__< sub > 1 </ sub > + … +

**a**< sub > k </ sub >

__X__< sub > k </ sub >

**is**normally distributed

**.**

__Every__significant section

**of**roadway maintained

**by**

**the**state

**is**assigned

**a**number

**,**officially State Highway Route

__X__but commonly called Route

__X__

**by**

**the**NJDOT and

**the**general public

**.**

__Every__variable

__X__< sub > i </ sub > in

**the**sequence

**is**associated with

**a**Bernoulli trial or experiment

**.**

__Every__time someone gave an answer that was not on

**the**board

**,**

**the**family lose

**a**life

**,**accompanied

**by**

**a**large "

__X__" on

**the**board with

**the**infamous " uh-uhh " sound

**.**

__Every__sigma-ideal on

__X__can be recovered in this way

**by**placing

**a**suitable measure on

__X__

**,**although

**the**measure may be rather pathological

**.**

*****

__Every__cover

**is**

**a**local homeomorphism — that

**is**

**,**for every

**,**there exists

**a**neighborhood

**of**c and

**a**neighborhood

**of**such that

**the**restriction

**of**p to U yields

**a**homeomorphism from U to V

**.**This implies that C and

__X__share all local properties

**.**

__Every__universal cover p: D →

__X__

**is**regular

**,**with deck transformation group being isomorphic to

**the**fundamental group

**.**

__Every__closed

**point**

**of**Hilb (

__X__) corresponds to

**a**closed subscheme

**of**

**a**fixed scheme

__X__

**,**and every closed subscheme

**is**represented

**by**such

**a**

**point**

**.**

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