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Every compact Lie group G of dimension > 0 has a subgroup isomorphic to the circle group.
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Every and compact
* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.
* 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.
* Sequentially compact: Every sequence has a convergent subsequence.
* Countably compact: Every countable open cover has a finite subcover.
* Limit point compact: Every infinite subset has an accumulation point.
Every compact metric space is complete, though complete spaces need not be compact.
Every entire function can be represented as a power series that converges uniformly on compact sets.
* Every compact metric space ( or metrizable space ) is separable.
* Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.
Every continuous function on a compact set is uniformly continuous.
Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space.
* Every compact Hausdorff space of weight at most ( see Aleph number ) is the continuous image of ( this does not need the continuum hypothesis, but is less interesting in its absence ).
*( BCT2 ) Every locally compact Hausdorff space is a Baire space.
Every group has a presentation, and in fact many different presentations ; a presentation is often the most compact way of describing the structure of the group.
*( BCT2 ) Every locally compact Hausdorff space is a Baire space.
Every H * is very special in structure: it is pure-injective ( also called algebraically compact ), which says more or less that solving equations in H * is relatively straightforward.
Every and Lie
* 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 homomorphism f: G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and.
Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.
* Every Lie algebra is a Lie algebroid over the one point manifold.
* Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.
* Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.
Consider a Lie algebra g over a field K. Every element x of g defines the adjoint endomorphism ad ( x ) ( also written as ad < sub > x </ sub >) of g with the help of the Lie bracket, as
Every quantum field theory satisfying certain technical assumptions about its S-matrix that has non-trivial interactions can only have a symmetry Lie algebra which is always a direct product of the Poincaré group and an internal group if there is a mass gap: no mixing between these two is possible.
The album features such hits as attitudized song " Every Little Lie " which tell a tale of an unfaithful man who instead of ending the relationship amicablely, decides to lead her on.
*" Every Little Lie "
Every left-invariant vector field on a Lie group is complete.
Every and group
Every finite simple group is isomorphic to one of the following groups:
Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.
* Every group G acts on G, i. e. in two natural but essentially different ways:, or.
Every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters.
Every instrumental group ( or section ) has a principal who is generally responsible for leading the group and playing orchestral solos.
* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.
While he admits the existence of caste-based discrimination, he writes that " Every social group cannot be regarded as a race simply because we want to protect it against prejudice and discrimination ".
Every local group is required to have a seneschal who reports to the kingdom's seneschal.
Every local group is required to have one.
* Every topological group is completely regular.
Every summer the group gathers in Newport, RI for week long dance training, seaside teas, and evenings enjoying the splendors of the Gilded Age.
* Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem
Every synset contains a group of synonymous words or collocations ( a collocation is a sequence of words that go together to form a specific meaning, such as " car pool "); different senses of a word are in different synsets.
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 group is naturally isomorphic to its opposite group "
Every ten years, when the general census of population takes place, each citizen has to declare which linguistic group they belong or want to be aggregated to.
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