Thus the films seen as they came in ( coordinated for the regular sections ), were often out of context.

Thus ℵ < sub > 0 </ sub > is a regular cardinal.

Thus the category of completely regular spaces CReg is a reflective subcategory of Top, the category of topological spaces.

Thus, John Dryden, among many others, compares the " irregular " Shakespeare with the " regular " Ben Jonson in his Essay of Dramatick Poesie ( 1668 ), and makes use of the unity of time in this passage criticizing Shakespeare's history plays:

Thus, X-ray diffraction results from an electromagnetic wave ( the X-ray ) impinging on a regular array of scatterers ( the repeating arrangement of atoms within the crystal ).

Thus, strict rules apply for entrance, including Church membership and regular attendance.

Thus, the definition of T < sub > 3 </ sub > may cite T < sub > 0 </ sub >, T < sub > 1 </ sub >, or T < sub > 2½ </ sub > instead of T < sub > 2 </ sub > ( Hausdorffness ); all are equivalent in the context of regular spaces.

Thus a regular space encountered in practice can usually be assumed to be T < sub > 3 </ sub >, by replacing the space with its Kolmogorov quotient.

Thus the behavior of the function on the Fatou set is ' regular ', while on the Julia set its behavior is ' chaotic '.

Thus, surfactant as an antiadhesive in the regular collapse and reopening of alveoli may be the primary function of the system in pinnipeds.

Thus chunks with odd lengths will be " padded " to an even byte boundary by adding a so-called " pad byte " after their regular end.

Thus, in the Majors, a " playoff " is actually part of the regular season and thus can be called a " Pennant playoff ".

Thus, as the vehicle goes through the regular bumping and dipping that comes with driving over a terrain, the circuit is completed and the bomb or explosive is allowed to function.

Thus, the transfer of the PC to the regular force of the new military organization was effected under the provisions of Sec.

Thus, although armored riders were used in the Roman army as early as the 2nd century BC ( Polybios, VI, 25, 3 ), the first recorded deployment and use of cataphracts ( equites cataphractarii ) by the Roman Empire comes in the 2nd century AD, during the reign of Emperor Hadrian ( 117-138 AD ), who created the first, regular unit of auxiliary, mailed cavalry called the ala I Gallorum et Pannoniorum catafractata.

Thus trade traffic could easily be controlled and income from tolls was frequent and regular.

Thus, it is not a violation of the PES if one delivers a letter of one's friend even without affixation of postage or if a company has one of its regular employees deliver mail that originates from the company to its customers.

Thus, employers also benefit as experienced interns need little or no training when they begin full-time regular employment.

Thus, regular script has parentage in early semi-cursive as well as neo-clerical scripts.

Thus on 1 May 1961 the TA divisional hedquarters were merged with regular army districts, which were matched with Civil Defence Regions to aid mobilisation for war.

Thus even " regular " mail may make part of its journey on an aircraft.

Thus " regular " 1861-O halves with this crack probably were used by the Confederates for some of the mass striking.

Thus on a regular basis, he quotes secular scientists in his Torah commentary.

Thus, what is known as the extreme value theorem in calculus generalizes to compact spaces.

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

Thus 2. 5 cm < sup > 3 </ sup > contain about a googol Planck spaces.

Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

) Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the Riemann mapping theorem provides a large group of conformal transformations.

Thus, manifolds, Lie groups, and graphs are all called connected if they are connected as topological spaces, and their components are the topological components.

Thus, when discussing simply connected topological spaces, it is far more common to speak of simple connectivity than simple connectedness.

Thus domes, like arches, have a great deal of structural strength when properly built and can span large open spaces without interior supports.

Thus, unitary operators are just automorphisms of Hilbert spaces, i. e., they preserve the structure ( in this case, the linear space structure, the inner product, and hence the topology ) of the space on which they act.

Thus a particular kind of map of spaces is well modelled by a particular kind of morphism of groupoids.

Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups.

The category Top of topological spaces has as morphisms the continuous functions, where a function f is continuous if the inverse image f < sup > − 1 </ sup >( O ) of any open set in the codomain of f is open in the domain of f. Thus any continuous function f from a space X to a space Y defines an inverse mapping f < sup > − 1 </ sup > from Ω ( Y ) to Ω ( X ).

Specifically, if X and Y are the representation spaces of two linear representations of G then a linear map f: X → Y is called an intertwiner of the representations if it commutes with the action of G. Thus an intertwiner is an equivariant map in the special case of two linear representations / actions.

Thus metric spaces and metric maps form a category Met ; Met is a subcategory of the category of metric spaces and Lipschitz functions, and the isomorphisms in Met are the isometries.

For the spaces, the P < sub > n </ sub > is reflexive if and only if n < p. Thus, no is polynomially reflexive.

Thus in characteristic 0, the symmetrization map is an isomorphism of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra.

Thus, as boundaries fade in importance due to transnational motion, the integration of global and local forces defined by glocalization make transnational social spaces, in which those people and processes that have crossed borders interact, a reality ; ultimately, this process of glocalization may provide societal encouragement for the more culturally open mindset of cosmopolitanism.

Thus, for a function between metric spaces it is equivalent to admit a modulus of continuity which is either concave, or subadditive, or uniformly continuous, or sublinear.

Thus, as is typical of urbeach changing facilities, only one sixth of the washroom space is devoted to use for changing, and the men's and women's washrooms / changerooms in total only comprise six parking spaces worth of space in P1.