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 locally compact topological field K, an absolute value can be defined as follows.

Given constants C, D and V, there are only finitely many ( up to diffeomorphism ) compact n-dimensional Riemannian manifolds with sectional curvature | K | ≤ C, diameter ≤ D and volume ≥ V.

Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.

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

is also true in the case of compact manifolds, due to Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold ( M, I ), it is always equipped with a compatible hyperkähler metric.

Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of and its analytic capacity, it might be

Given any compact Lie group G one can take its identity component G < sub > 0 </ sub >, which is connected.

Given a Morse function on a compact boundaryless manifold M, such that the critical points of f satisfy < math > f ( p_1 ) < f ( p_2 ) <

Given the Rover's equipment, prices were reasonably competitive in the large family car segment and considerably lower than the price of such compact executive cars such as the BMW 3 Series and Audi A4.

Given a 3-manifold ( not necessarily compact ) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups.

Given a dual pair with a locally convex space and its continuous dual then is a dual topology on if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of

Given a dual pair with a topological vector space and its continuous dual the Mackey topology is a polar topology defined on by using the set of all absolutely convex and weakly compact sets in.

Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first strongly compact cardinal is supercompact ; these cannot both be true, however.

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 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 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 two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

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:

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

* Given any morphism k ′: K ′ → X such that f k ′ is the zero morphism, there is a unique morphism u: K ′ → K such that k u

Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology.

Given a set S with three subsets, J, K, and L, the following holds:

Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is automatic.

Given a separable extension K ′ of K, a Galois closure L of K ′ is a type of splitting field, and also a Galois extension of K containing K ′ that is minimal, in an obvious sense.