Although not part of this definition, many authors require that the topology on G be Hausdorff ; this corresponds to the identity map being a closed inclusion ( hence also a cofibration ).

Although four-and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done.

Although he usually designed entirely by hand, Pei used a computer to " confirm the spaces " for the Morton H. Meyerson Symphony Center.

Although their form was variable, basilicas often contained interior colonnades that divided the space, giving aisles or arcaded spaces on one or both sides, with an apse at one end ( or less often at each end ), where the magistrates sat, often on a slightly raised dais.

Although we seemed to have little in common — I was a redneck from Aragon, and he an elegant Andalusian — we spent most of our time together ... We used to sit on the grass in the evenings behind the Residencia ( at that time, there were vast open spaces reaching to the horizon ), and he would read me his poems.

Although opinions differ as to which situations are objectionable, some feminists see objectification of women taking place in the sexually oriented depictions of women in advertising and media, women being portrayed as weak or submissive through pornography, images in more mainstream media such as advertising and art, stripping and prostitution, men brazenly evaluating or judging women sexually or aesthetically in public spaces, and the presumed need for cosmetic surgery, particularly breast enlargement and labiaplasty.

Although that is strictly speaking a question about a real vector bundle ( the " hairs " on a ball are actually copies of the real line ), there are generalizations in which the hairs are complex ( see the example of the complex hairy ball theorem below ), or for 1-dimensional projective spaces over many other fields.

Although architectural acoustics was first applied to opera houses and concert halls, this branch of acoustical engineering applies to any enclosed area, whether concert halls, office spaces, or ventilation ducts.

Although it still serves as a dormitory, it now also houses dance studios, meeting spaces, and administrative offices.

Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications.

Although Loughton is surrounded by open countryside it contains in addition many parks and open spaces.

Although some ( co ) homology theories are computable using tools of linear algebra, many other important ( co ) homology theories, especially singular ( co ) homology, are not computable directly from their definition for nontrivial spaces.

Although Polish spaces are metrizable, they are not in and of themselves metric spaces ; each Polish space admits many complete metrics giving rise to the same topology, but no one of these is singled out or distinguished.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry.

Although rg chromaticity contains less information than RGB or HSV color spaces, it has a number of useful properties for computer vision applications.

Although Toronto has often been criticized for not having a dynamic and beautiful waterfront park, harbourfront has a network of parks, open spaces and trails that allow residents and visitors to access the public realm.

Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn – Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Although subpixel rendering increases the number of reconstruction points on the display this does not always mean that higher resolution, higher spatial frequencies, more lines and spaces, may be displayed on a given arrangement of colour subpixels.

Although Metro reserved on 15 percent of all spaces for such parking at most stations, at Anacostia this was raised to 25 percent.

Although considered a built-up urban area, Archway has a number of parks and green spaces, no fewer than nine within a mile of the tube station: Archway Park, Hillrise Park, Dartmouth Park, Elthorne Park, Sunnyside Gardens, Crouch Hill Park, Waterlow Park, Whittington Park, Foxham Gardens, and Tufnell Park Playing Fields.

Although not a very prolific scorer, Vryzas ' tenacious work ethic, his restless battles with the opposing defenders, his talent to create chances and open spaces for his attacking partners, in addition to his aerial ability and passing skills made him a valuable member of all the Clubs he played for, as well as the Greek National Team with which he won the European Championship in 2004.

Although supermanifolds are special cases of noncommutative manifolds, the local structure of supermanifolds make them better suited to study with the tools of standard differential geometry and locally ringed spaces.

Although Perkins was also performing with Beasts of Bourbon, The Cruel Sea built a following on the inner-city pub rock circuit with " atmospheric music evoked the feel of wide open spaces ".

Although the terminology of vector spaces is used below and the symbol R can be taken to mean the field of real numbers, the results discussed hold whenever R is a commutative ring and vector space is everywhere replaced with free R-module.

Although they aren't included in the regular CD / DVD shipments, subscribers can download old software such as MS-DOS 5. 0 and Windows 3. 1 from MSDN Subscriber Downloads.

Although San Antonio had over 650, 000 people at the time ( and has since grown to become the seventh-largest city in the United States ), it has always been a medium-sized market because the surrounding suburban and rural areas aren't much larger than the city itself.

Although Koopa Troopa was replaced by Wario on the roster of 1996's Mario Kart 64, a friendly Koopa was given the role of the host of 1998's Mario Party for the Nintendo 64, proving that Koopa Troopas aren't necessarily all loyal to Bowser.

" He also said, " Although Hootie & the Blowfish aren't innovative, they deliver the goods, turning out an album of solid, rootsy folk-rock songs that have simple, powerful hooks.

Although all three shorts use sound, the voice actors aren't credited and were never revealed.

Although bit of a misnomer as the boats aren't without limits, unlimited hydroplanes have much fewer restrictions than with limited hydroplane racing.

Although seaplanes aren't quite as popular anymore, the basin is still there at Davis Islands

" Although the two aren't really similar, it brings to mind the band's hit, " TV Dinners.

Although this may be a valid and accepted practice in real-life sports, in online games stacking upsets less-skilled players who feel that they aren't being given a fair chance.

Although specific details aren't clear right now, it may be the extension from Yufa into Gu ' an in Hebei province, which currently is part of China National Highway 106, but which many maps claim now to be already expressway.

Although the SSY is affiliated to the Scottish Socialist Party, it is an autonomous body and is open to all socialists ( who aren't members of opposing parties ) under 27 living in Scotland.

Although CODAN is by no means original or unique technology or a unique product, it has achieved recognition in the security market that exclusively digital methods aren't always needed.

Although the shows aren't spin-offs in the traditional sense, Sunset was the first in this chain of " exotic location detective series ".

Although most black eye injuries aren't serious, bleeding within the eye, called a hyphema, is serious and can reduce vision and damage the cornea.

Although legal tender, the notes aren't intended for circulation.

Although they aren't biologically related, Vicki cannot accept it and rebels by dating Ash Ferreira ( Raji James ) briefly.

Although it is generally harmless to plant growth in pH-neutral soils, the concentration in acid soils of toxic Al < sup > 3 +</ sup > cations increases and disturbs root growth and function.

Although Rodin is generally considered the progenitor of modern sculpture, he did not set out to rebel against the past.

Although the early German Panzerjäger carried more effective weapons than the tanks on which they were based, they were generally lacking in protection for the crew, having thinly armored open-topped superstructures.

Although they are generally classified as a cappella metal, the band also includes a drummer, and use amplifiers on some songs to distort the voice to sound more like a real guitar.

Although reactions to any drug may vary from person to person, generally those medications that contain aluminum or calcium are the likeliest to cause constipation, those that contain magnesium are the likeliest to cause diarrhea.

Although influential for a time their popularity did not generally survive the horrors of ideological warfare, as the competing ideologies utilized their systems in their platforms.

Although the term “ biochemistry ” seems to have been first used in 1882, it is generally accepted that the formal coinage of biochemistry occurred in 1903 by Carl Neuberg, a German chemist.

Although the Iron Age generally followed the Bronze Age, in some areas, the Iron Age intruded directly on the Neolithic from outside the region except for Sub-Saharan Africa where it was developed independently.

" Although many evangelicals allow that conversion can be a process, generally they see it as a specific, identifiable moment of time when a person simply and sincerely trusts in Jesus Christ as savior.

Although earlier precursors to this presentation exist, its invention is generally credited to Russian chemist Dmitri Mendeleev in 1869, who intended the table to illustrate recurring trends in the properties of the elements.

Although central banks today are generally associated with fiat money, the 19th and early 20th centuries central banks in most of Europe and Japan developed under the international gold standard, elsewhere free banking or currency boards were more usual at this time.

Although the word " chalcogen " is literally taken from Greek words being " copper-former ", the meaning is more in line with " copper-ore former " or more generally, " ore-former ".

Although this selected translation is not generally quoted by members of the Church, the English Bible issued by the Church and commonly used by Latter-day Saints contains cross references to the Joseph Smith Translation ( JST ), as well as an appendix containing major excerpts from it.

Although Cetus is not generally considered part of the zodiac, the ecliptic passes close to its constellation boundary, and thus the planets may be seen in Cetus for brief periods of time.

Although the public generally associates carcinogenicity with synthetic chemicals, it is equally likely to arise in both natural and synthetic substances.

Although rules for the high school, college, and NFL games are generally consistent, there are several minor differences.

Although never formally endorsed by CND nationally, INDEC candidates were generally put up by local CND groups as a means of raising the profile of this issue in public politics.

Although any music which uses computers in its composition or realisation is computer-generated to some extent, the use of computers is now so widespread ( in the editing of pop songs, for instance ) that the phrase computer-generated music is generally used to mean a kind of music which could not have been created without the use of computers.

Although Christians generally believe their religion to be very inclusive ( since not only Jews but all gentiles can be Christian ), Jews see Christianity as highly exclusive, because some denominations view non-Christians ( such as Jews and Pagans ) as having an incomplete or imperfect relationship with God, and therefore excluded from grace, salvation, heaven, or eternal life.

Although conservatism in politics was strong during the 1950s and Eisenhower generally shared these sentiments, his administration concerned itself mostly with foreign affairs ( an area that the career military president was more knowledgeable about ) and pursued a hands-off domestic policy.

Although the dachshund is generally an energetic dog, some are sedate.

Although further persecutionary edicts followed, compelling the arrest of the Christian clergy and universal acts of sacrifice, the persecutionary edicts were ultimately unsuccessful ; most Christians escaped punishment, and pagans too were generally unsympathetic to the persecution.

Although the very act of codification was a radical innovation, given the precedent-based design of the Roman legal system, the jurists were generally conservative, and constantly looked to past Roman practice and theory for guidance.

Although the existence of dark matter is generally accepted by the mainstream scientific community, several alternative theories have been proposed to try to explain the anomalies for which dark matter is intended to account.

" Although the term destroyer had been used interchangeably with the terms " TBD " and " torpedo boat destroyer " by navies since 1892, the term torpedo boat destroyer had been generally shortened to simply " destroyer " by nearly all navies by the First World War.