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Page "Totally bounded space" ¶ 0
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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 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 totally
Every non-empty totally ordered set is directed.
* Every totally ordered set with the order topology is Tychonoff.
* Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.
* Every totally ordered set is a distributive lattice with max as join and min as meet.
Every November the Reebok Stadium hosts Kidz up North which is one of the largest free UK exhibitions totally dedicated to children with disabilities and special needs, their parents, carers and professionals who work with them.
* Every compact set is totally bounded, whenever the concept is defined.
* Every totally bounded metric space is bounded.
Every compact metric space is totally bounded.

Every and bounded
* Pseudocompact: Every real-valued continuous function on the space is bounded.
Every finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation.
* Every continuous function f: → R is bounded.
This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded.
For example, to study the theorem “ Every bounded sequence of real numbers has a supremum ” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.
Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.
Every three years the Company makes an award to the three buildings or structures, in the area bounded by the M25 motorway, which respectively embody the most outstanding example of brickwork, of slated or tiled roof and of hard-surface tiled wall and / or floor.
Every Polish town was bounded to put up a quantity of soldiers-this was a conspicuous sign of a power of a given town how much soldiers it had to put up.
Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element.
Every bounded linear transformation from a normed vector space to a complete, normed vector space can be uniquely extended to a bounded linear transformation from the completion of to.
Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.
* Every finite set of points is bounded
* Every relatively compact set in a topological vector space is bounded.
Every topological vector space X gives a bornology on X by defining a subset to be bounded iff for all open sets containing zero there exists a with.
Every ( bounded ) convex polytope is the image of a simplex, as every point is a convex combination of the ( finitely many ) vertices.

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