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Every compact metric space is complete, though complete spaces need not be compact.

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## Some Related Sentences

Every and compact

*

__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__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__entire function can

**be**represented as a power series that converges uniformly on

__compact__sets

**.**

*

__Every__locally__compact__regular**space****is**completely regular**,**and therefore every locally__compact__Hausdorff**space****is**Tychonoff**.**__Every__

__compact__Hausdorff

**space**

**is**also locally

__compact__

**,**and many examples of

__compact__

**spaces**may

**be**found in the article

__compact__

**space**

**.**

*

__Every____compact__Hausdorff**space**of weight at most ( see Aleph number )**is**the continuous image of ( this does**not****need**the continuum hypothesis**,**but**is**less interesting in its absence ).__Every__group has a presentation

**,**and in fact many different presentations ; a presentation

**is**often the most

__compact__way of describing the structure of the group

**.**

__Every__H *

**is**very special in structure: it

**is**pure-injective ( also called algebraically

__compact__), which says more or less that solving equations in H *

**is**relatively straightforward

**.**

Every and metric

*

__Every__separable__metric__**space****is**isometric to a subset of the ( non-separable ) Banach**space**l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this**is**known as the Fréchet embedding**.**
*

__Every__separable__metric__**space****is**isometric to a subset of C (), the separable Banach**space**of continuous functions → R**,**with the supremum norm**.**__Every__smooth submanifold of R < sup > n </ sup > has an induced Riemannian

__metric__g: the inner product on each tangent

**space**

**is**the restriction of the inner product on R < sup > n </ sup >.

__Every__building has a canonical length

__metric__inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert

**space**

**.**

*

__Every__totally disconnected**compact**__metric__**space****is**homeomorphic to a subset of a countable product of discrete**spaces****.**
This

**is**really a special case of a more general fact:__Every__continuous function from a**compact****space**into a__metric__**space****is**bounded**.**
#

__Every__Riemannian__metric__on a Riemann surface**is**Kähler**,**since the condition for ω to**be**closed**is**trivial in 2 ( real ) dimensions**.**__Every__hyperkähler manifold M has a 2-sphere of complex structures ( i

**.**e

**.**integrable almost complex structures ) with respect to which the

__metric__

**is**Kähler

**.**

__Every__special uniformly continuous real-valued function defined on the

__metric__

**space**

**is**uniformly approximable by means of Lipschitz functions

**.**

__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__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

**.**

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