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* σ-compact space: there exists a countable cover by compact spaces
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σ-compact and space
Let X be a locally compact Hausdorff space equipped with a finite Radon measure μ, and let Y be a σ-compact Hausdorff space with a σ-finite Radon measure ρ.
In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.
* Every compact space is σ-compact, and every σ-compact space is Lindelöf ( i. e. every open cover has a countable subcover ).
The reverse implications do not hold, for example, standard Euclidean space ( R < sup > n </ sup >) is σ-compact but not compact, and the lower limit topology on the real line is Lindelöf but not σ-compact.
Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also a Baire space, then G is locally compact.
* The previous property implies for instance that R < sup > ω </ sup > is not σ-compact: if it were σ-compact, it would necessarily be locally compact since R < sup > ω </ sup > is a topological group that is also a Baire space.
The converse, however, is not true ; for example, the space of rationals, with the usual topology, is σ-compact but not hemicompact.
* sigma-algebras, sigma-fields and sigma-finiteness in measure theory ; more generally, the symbol σ serves as a shorthand for " countably ", e. g. a σ-compact topological space is one that can be written as a countable union of compact subsets.
Let G be a σ-compact, locally compact topological group and π: G U ( H ) a unitary representation of G on a ( complex ) Hilbert space H. If ε > 0 and K is a compact subset of G, then a unit vector ξ in H is called an ( ε, K )- invariant vector if π ( g ) ξ-ξ < ε for all g in K.
σ-compact and countable
σ-compact and compact
* Open mapping theorem ( topological groups ) states that a surjective continuous homomorphism of a locally compact Hausdorff group G onto a locally compact Hausdorff group H is an open mapping if G is σ-compact.
The fact that ( 5 ) implies ( 4 ) requires us to assume that G is σ-compact ( and locally compact ) ( Bekka et al., Theorem 2. 12. 4 ).
σ-compact and spaces
The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.
space and there
But there is hope, for Conservation Commissioner Bontempo has tagged the sanctuary as the kind of place the state hopes to include in its program to double its park space.
In general, such apartments afford more protection than smaller buildings because their walls are thick and there is more space.
Since there is a continual loss of micrometeoritic material in space because of the radiation effects, there must be a continual replenishment: otherwise, micrometeorites would have disappeared from interplanetary space.
Of course, if there is a dust blanket around the Earth, the fluxes in interplanetary space should be less than the figures given here.
Second, even if the characteristic polynomial factors completely over F into a product of polynomials of degree 1, there may not be enough characteristic vectors for T to span the space V.
If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.
to this he glued and fitted other pieces of paper and four taut strings, thus creating a sequence of flat surfaces in real and sculptural space to which there clung only the vestige of a picture plane.
Maggie was looking much happier already, clearing a space on the table and chattering about how she could put up a typewriter right there, and be brushing up on her typing so Eugenia wouldn't be ashamed of it.
Between the two countries, there are political disputes over several aspects of political control over the Aegean space, including the size of territorial waters, air control and the delimitation of economic rights to the continental shelf.
Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only be one such model.
His report of life there covers a wide range of topics, such as marriage in heaven ( where all angels are married ), children in heaven ( where they are raised by angel parents ), time and space in heaven ( there are none ), the after-death awakening process in the World of Spirits ( a place halfway between Heaven and Hell and where people first wake up after death ), the allowance of a free will choice between Heaven or Hell ( as opposed to being sent to either one by God ), the eternity of Hell ( one could leave but would never want to ), and that all angels or devils were once people on earth.
By the beginning of World War II, many towns and cities had built space, and there were numerous qualified pilots available.
Since there are infinitely many vectors in the basis, this is an infinite-dimensional Hilbert space.
Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.
By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.
This has to be planned in advance as if there is no such space partners have to bid a usually unmakeable slam contract.
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