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Given a topological space and a subset of, the subspace topology on is defined by
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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 topological space X, let G < sub > 0 </ sub > be the set X.
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 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 subset
Given a set of integers, does some nonempty subset of them sum to 0?
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 a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.
: Given any set A, there is a set such that, given any set B, B is a member of if and only if B is a subset of A.
Given a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).
Given a set of integers, FIND-SUBSET-SUM is the problem of finding some nonempty subset of the integers that adds up to zero ( or returning the empty set if there is no such subset ).
Given a set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero.
Given any subset F =
Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).
* Rural postman problem: Given is also a subset of the edges.
for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:
Given a subset of the index set, the partial hypergraph generated by is the hypergraph
Given a subset, the section hypergraph is the partial hypergraph
Given a subset V of A < sup > n </ sup >, we define I ( V ) to be the ideal of all functions vanishing on V:
Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.
Given a ring R and a subset S, one wants to construct some ring R * and ring homomorphism from R to R *, such that the image of S consists of units ( invertible elements ) in R *.
; Generating set: Given a field extension E / F and a subset S of E, we write F ( S ) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +,-,*,/ on the elements of F and S. If E = F ( S ) we say that E is generated by S over F.
Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.
Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?
Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).
Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of and its analytic capacity, it might be
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