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Some Related Sentences
Every and subset
Every such
subset has
a smallest element, so to specify our choice function we can simply say that it maps each
set to
the least element
of that
set.

** Zorn's lemma:
Every non-empty partially ordered
set in which every chain
( i
. e
. totally ordered
subset ) has an upper bound contains at least one maximal element
.

The restricted principle "
Every partially ordered
set has
a maximal totally ordered
subset "
is also equivalent to AC over ZF
.

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

#
Every infinite
subset of X has
a complete accumulation point
.

#
Every infinite
subset of A has at least one limit point in
A.

* Limit point compact:
Every infinite
subset has an accumulation point
.

*
Every cofinal
subset of a partially ordered
set must contain
all maximal elements
of that
set.

*
Every separable metric
space is homeomorphic to
a subset of the Hilbert cube
.

*
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 element s, except
a possible greatest element, has
a unique successor
( next element
), namely the least element
of the subset of all elements greater than s
. Every subset which has an upper bound has
a least upper bound
.
Every subset of a nowhere dense
set is nowhere dense, and
the union
of finitely many nowhere dense
sets is nowhere dense
.

*
Every subset of Baire
space or Cantor
space is an open
set in
the usual topology on
the space.

*
Every arithmetical
subset of Cantor
space of < sup >( or?
Every subset of the Hilbert cube inherits from
the Hilbert cube
the properties
of being both metrizable
( and therefore T4 ) and second countable
.

It
is more interesting that
the converse also holds:
Every second countable T4
space is homeomorphic to
a subset of the Hilbert cube
.

*
Every totally disconnected compact metric
space is homeomorphic to
a subset of a countable product
of discrete spaces
.

*
Every irreducible closed
subset of P < sup > n </ sup >( k )
of codimension one
is a hypersurface ; i
. e.,
the zero
set of some homogeneous polynomial
.

*
Every finite or cofinite
subset of the natural numbers
is computable
.
Every and vector

**
Every vector space has
a basis
.
Every vector v in determines
a linear map from R to taking 1 to v, which can be thought
of as
a Lie algebra homomorphism
.
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 finite-dimensional
vector space is isomorphic to its dual
space, but this isomorphism relies on an arbitrary choice
of isomorphism
( for example, via choosing
a basis and then taking
the isomorphism sending this basis to
the corresponding dual basis ).
Every random
vector gives rise to
a probability measure on R < sup > n </ sup > with
the Borel algebra as
the underlying sigma-algebra
.
Every continuous function in
the function
space can be represented as
a linear combination
of basis functions, just as every
vector in
a vector space can be represented as
a linear combination
of basis vectors
.
Every vector in
the space may be written as
a linear combination
of unit vectors
.
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 coalgebra, by
( vector space ) duality, gives rise to an algebra, but not in general
the other way
.

*
Every holomorphic
vector bundle on
a projective variety
is induced by
a unique algebraic
vector bundle
.

For example, second-order arithmetic can express
the principle "
Every countable
vector space has
a basis " but it cannot express
the principle "
Every vector space has
a basis ".

Many principles that imply
the axiom
of choice in their general form
( such as "
Every vector space has
a basis ") become provable in weak subsystems
of second-order arithmetic when they are restricted
.
Every vector space is free, and
the free
vector space on
a set is a special case
of a free module on
a set.
Every and space

**
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 compact metric
space is separable
.

*
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 compact metric
space is complete, though complete spaces need not be compact
.
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 )
0.174 seconds.