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Given a topological space X, let G < sub > 0 </ sub > be the set X.
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Given and topological
* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.
Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.
Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).
Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.
Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.
Given any topological space X, the zero sets form the base for the closed sets of some topology on X.
Given a locally compact topological field K, an absolute value can be defined as follows.
Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.
Given a complex vector bundle V over a topological space X,
Given a point x of a topological space X, and two maps f, g: X → Y ( where Y is any set ), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal ;
Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module.
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.
Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain ( also called fundamental region ) for this action is a set D of representatives for the orbits.
Given a compact topological space X, the topological K-theory K < sup > top </ sup >( X ) of ( real ) vector bundles over X coincides with K < sub > 0 </ sub > of the ring of continuous real-valued functions on X.
Given a topological group G, the Bohr compactification of G is a compact Hausdorff topological group Bohr ( G ) and a continuous homomorphism
Given any topological space X we can define a ( possibly ) finer topology on X which is compactly generated.
Given a topological space and a subset of, the subspace topology on is defined by
Given a topological space X = 〈 X, T 〉 one can form the power set Boolean algebra of X:
Given a field of sets the complexes form a base for a topology, we denote the corresponding topological space by.
Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interior and closure.
Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra ( which form a base for a topology ).
Given and space
* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.
Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).
Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).
Given a vector space V over the field R of real numbers, a function is called sublinear if
Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.
Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space
Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.
Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.
Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and
Given an operator on Hilbert space, consider the orbit of a point under the iterates of.
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.
Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.
Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.
Given a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X.
Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:
Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system
Given and X
: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).
Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).
* Given any set X, there is an equivalence relation over the set of all possible functions X → X.
Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as
# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )
Given X such that
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