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Page "Compact space" ¶ 55
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Every and topological
* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.
* Every topological group is completely regular.
Every group can be trivially made into a topological group by considering it with the discrete topology ; such groups are called discrete groups.
Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.
Every subgroup of a topological group is itself a topological group when given the subspace topology.
Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.
Every local field is isomorphic ( as a topological field ) to one of the following:
* 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 directed acyclic graph has a topological ordering, an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge.
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 Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological group.
Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.
* Every constant function between topological spaces is continuous.
Every topological group is an H-space ; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.
Every interior algebra can be represented as a topological field of sets with its interior and closure operators corresponding to those of the topological space.
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 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 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 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 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 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.

Every and X
# Every net on X has a convergent subnet ( see the article on nets for a proof ).
# Every filter on X has a convergent refinement.
# Every ultrafilter on X converges to at least one point.
# Every infinite subset of X has a complete accumulation point.
Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).
* Every linear combination of its components Y = a < sub > 1 </ sub > X < sub > 1 </ sub > + … + a < sub > k </ sub > X < sub > k </ sub > is normally distributed.
Every significant section of roadway maintained by the state is assigned a number, officially State Highway Route X but commonly called Route X by the NJDOT and the general public.
Every variable X < sub > i </ sub > in the sequence is associated with a Bernoulli trial or experiment.
* Every X is a Y.
Every Gauss – Markov process X ( t ) possesses the three following properties:
Every time someone gave an answer that was not on the board, the family lose a life, accompanied by a large " X " on the board with the infamous " uh-uhh " sound.
Every sigma-ideal on X can be recovered in this way by placing a suitable measure on X, although the measure may be rather pathological.
* Every cover is a local homeomorphism — that is, for every, there exists a neighborhood of c and a neighborhood of such that the restriction of p to U yields a homeomorphism from U to V. This implies that C and X share all local properties.
Every universal cover p: D → X is regular, with deck transformation group being isomorphic to the fundamental group.
Every cumulant is just μ times the corresponding cumulant of the constant random variable X = 1.
Every closed point of Hilb ( X ) corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.

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