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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 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 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 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 Baire

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

*( BCT2 )
Every locally compact Hausdorff
space is a
Baire space.

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

*( BCT2 )
Every locally compact Hausdorff
space is a
Baire space.
* 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 open subspace
of a
Baire space is a
Baire space.
* Every Polish
space is obtained as a continuous image
of Baire space ;
in fact every Polish
space is the image
of a continuous bijection defined
on a closed
subset of Baire space.
* Every set of reals
in L ( R )
is Lebesgue measurable (
in fact, universally measurable ) and has
the property
of Baire and
the perfect
set property
.
Every and space

**
Every vector
space has a basis
.

**
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 )
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
.
0.114 seconds.