Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.

Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.

Since coverings are local homeomorphisms, a covering of a topological n-manifold is an n-manifold.

Since the topologists ' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.

Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups.

Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete.

Since multiplication and inversion are continuous functions on C < sup >×</ sup >, the circle group has the structure of a topological group.

Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a topological ring.

Since there are no face-transitive polyhedra with five sides and there are two distinct topological types, this term is less frequently used than tetrahedron or octahedron.

Since the morphisms are generated by those that ' skip ' or ' add ' a single element, the detailed relations written out above underlie the topological applications.

Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the weak homotopy equivalences, the theorem can equivalently be stated for functors on a category defined in this way.

As with all covering spaces, the fundamental group of G injects into the fundamental group of H. If G is path-connected then the quotient group is isomorphic to K. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space.

Since uniform spaces come as topological spaces and uniform isomorphisms are homeomorphisms, every topological property of a uniform space is also a uniform property.

Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold.

Since the demise of Commodore, various groups have marketed successors to the original Amiga line, including Genesi, Eyetech, ACube Systems and A-EON Technology, and AmigaOS has influenced replacements, clones and compatible systems such as MorphOS, AmigaOS 4 and AROS.

Since then groups that are not affiliated to the Santo Daime have used ayahuasca, and a number of different ' styles ' have been developed, such as the non-religious approach developed by Daniel Waterman in 2001, popularly termed Ayahuasca Open Style ( AOS ).

Since its original 1973 Broadway production, the musical has enjoyed professional productions in the West End, by opera companies, in a 2009 Broadway revival, and elsewhere, and it is a popular choice for regional groups.

Since that time, efforts to use biological warfare has been more apparent in small radical organizations attempting to create fear in the eyes of large groups.

Chaplin decided not to re-enter the United States, writing: " Since the end of the last world war, I have been the object of lies and propaganda by powerful reactionary groups who, by their influence and by the aid of America's yellow press, have created an unhealthy atmosphere in which liberal-minded individuals can be singled out and persecuted.

Since the independence in 1975 the official statistics in Cape Verde have no longer made statistics based on racial groups.

Since the reunions in the UK and Australia in 1957, two generations of Christadelphians have grown up with little awareness of the existence of the minority " fellowships ", or awareness that the main group is called " Central " by the minority groups.

Since the 1980s theologians from the Oriental ( non-Chalcedonian ) Orthodox and Eastern ( Chalcedonian ) Orthodox churches have been meeting in a bid to resolve theological differences, and have concluded that many of the differences are caused by the two groups using different terminology to describe the same thing ( see Agreed Official Statements on Christology with the Eastern Orthodox Churches ).

Since at least the 1940s, the approach of traditional Christians was to apply the meaning of cult such that it included those religious groups who use other scriptures beside the Bible or have teachings and practices deviating from traditional Christian teachings and practices.

Since then differences between the Hasidim and their opponents have slowly diminished and both groups are now considered part of Haredi Judaism.

Since the Marxist revolution in 1978, the city has been a target of foreign-backed militant groups such as the Mujahideen, Taliban, Haqqani network, Hezbi Islami, and others.

Since the rebranding in 2010, Sporting have seen dramatic growth in their fan section, with several fan groups adding their voice to The Cauldron culture and atmosphere.

Since 1999, the education laws have forbidden the public universities to instruct students in languages other than Latvian ( there are exclusions made for linguistics, some international projects and non-budget groups ).

Since the 1970s the city has also been home to a rich, if fluctuating, array of independent theatre groups and some show / musical companies.

Since the enactment of EU legislation concerning the concentration of particulate in the air, environmental groups such as Greenpeace have staged large protest rallies to urge the city council and the State government to take a harder stance on pollution.

Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.

Since all people including the most isolated indigenous groups such as the Andamanese or the Tasmanian aboriginals possess language, then it was presumedly present in the ancestral populations in Africa before the human population split into various groups to inhabit the rest of the world.

Since the 1990s, there have been organized Hellenic groups practising in Greece.

Since then, it has become clear that the Plantae as originally defined included several unrelated groups, and the fungi and several groups of algae were removed to new kingdoms.

Since the Chechen separatists had declared independence in the early 1990s, an intermittent guerrilla war was fought between the rebel groups and the Russian military.

Since the state controls the means of coercion, and given the dependence of many groups in civil society on the state for achieving any goals they may espouse, state personnel can to some extent impose their own preferences on civil society.

Since the early 1990s, the legitimatization of the principle of national self-determination has led to an increase in the number of conflicts within states, as sub-groups seek greater self-determination and even full secession, and as their conflicts for leadership within groups and with other groups and with the dominant state become violent.

Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Since we can neither undo nor redo the past, we are limited to the events of today and tomorrow.

Since the recognition of Israel as a nation state, claims are made in many cases which identify the claimant as a member of the new body politic.

Since accurate base maps are necessary for any planning program, the first step taken by the planning division to implement the long-range state plan has been to prepare two series of base maps -- one at a scale of 1 inch to a mile, and the second a series of 26 sheets at a scale of 1 inch to 2000 feet, covering the entire state.

Since conventional methods are insensitive at the low frequencies of these molecular transitions, the paramagnetic resonance method is being used instead.

Since broadcast frequencies are very limited in number, these objectives are to some extent inconsistent in that not all of them can be fully realized, and to the extent that each is realized, there is a corresponding reduction of the possibilities for fullest achievement of the others.

Since the provision of skywave service requires adequate freedom from interference, only class 1, stations are capable of rendering skywave service.

Since the sides are also covered up to the spray rails, they are also rough-sanded in that area.

Since fly cutters are one sided and not balanced, they should be used at the slowest speed available, and fed very slowly to avoid binding.

Since there are almost no schools in the country offering graduate work in interior design this rule cannot at present be observed.

Since charges are relatively the same, reserving a car before you leave for Europe will assure you of having one on tap when you want it.

Since practical washing procedures are both lengthy and expensive, a number of laboratory tests have been developed for the numerical evaluation of detergents.

Since the circulating thyroid hormones are the amino acids thyroxine and tri-iodothyronine ( cf. Section C ), it is clear that some mechanism must exist in the thyroid gland for their release from proteins before secretion.

Since emotional reactions in the higher vertebrates depend on individual experience and are aroused in man, in addition, by complex symbols, one would expect that the hypothalamus could be excited from the cortex.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

Since D and D' are both diagonalizable and they commute, they are simultaneously diagonalizable, and Af is diagonalizable.

Since N and N' are both nilpotent and they commute, the operator Af is nilpotent ; ;

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

Since the choice is by lot each week, the outcomes of different trials are independent.

Since two curves of symbol Af on Q intersect in Af points, it follows that there are Af lines of Af which are tangent to Aj.