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Some Related Sentences
Every and separable
* Every compact
metric space is separable.
* Every compact
metric space ( or metrizable
space )
is separable.
* 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 separable metric space is isometric
to a subset of the
* Every second-countable
space is first-countable,
separable, and Lindelöf
.
* Every subextension
of F / k
is separable.
* Every finite subextension
of F / k
is separable.
* Every polynomial over k
is separable.
* Every finite extension
of k
is separable.
* Every algebraic extension
of k
is separable.
* Every reduced commutative k-algebra A
is a separable algebra ; i
. e.,
is reduced for every field extension F / k
.
Every commutative von Neumann algebra on
a separable Hilbert space is isomorphic
to L < sup >∞</ sup >( X ) for some standard measure
space ( X, μ ) and conversely, for every standard measure
space X, L < sup >∞</ sup >( X )
is a von Neumann algebra
.
Every separable topological
space is ccc
.
Every metric space which
is ccc
is also
separable, but in general
a ccc topological
space need not be
separable.
Every second-countable manifold
is separable and paracompact
.
Every and metric
Every compact
metric space is complete, though complete spaces need not be compact
.
* Every metric space is Tychonoff ; every pseudometric
space is completely regular
.
Every uniformly continuous function between
metric spaces
is continuous
.

*( BCT1 )
Every complete
metric space is a Baire
space.
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 >.

*( BCT1 )
Every complete
metric space is a Baire
space.
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 totally bounded
metric space is bounded
.
Every compact
metric space is totally bounded
.
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 ( almost ) complex manifold admits
a Hermitian
metric.
Every and space

**
Every vector
space has
a basis
.

**
Every infinite game in which
is a Borel
subset of Baire
space is determined
.

**
Every Tychonoff
space has
a Stone – Čech compactification
.
* Theorem
Every reflexive normed
space is a Banach
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 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
.
* Pseudocompact:
Every real-valued continuous function on
the space is bounded
.
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 point in three-dimensional Euclidean
space is determined by three coordinates
.
Every node on
the Freenet network contributes storage
space to hold files, and bandwidth that it uses
to route requests from its peers
.
Every space filling curve hits some points multiple times, and does not have
a continuous inverse
.
* 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
.
0.447 seconds.