However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis.

However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.

However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and in this generality only the latter property is used to define compactness.

However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using ( BPIT ), and in fact the converse also holds.

However the sensitivity of the fiber gyro is enhanced by having a long optical fiber coiled for compactness, but in which the Sagnac effect is multiplied according to the number of turns.

However, by the extreme value theorem ( or the classical compactness argument ), a continuous function on a compact interval has a maximum, and thus an arg max.

However, in granting nationalities union republic status, three additional factors were considered: a population of at least 1 million, territorial compactness, and location on the borders of the Soviet Union.

However, Ice might not offer similar performance or compactness advantages when SOAP messages are exchanged using a more efficient transport and message encoding such as SOAP / TCP or Fast Infoset.

However, there is a minimum of legislative restriction on boating.

However, for backward compatibility with early HTML authors and browsers that ignored this restriction, raw characters and numeric character references in the 80 – 9F range are interpreted by some browsers as representing the characters mapped to bytes 80 – 9F in the Windows-1252 encoding.

However, this problem has been alleviated somewhat by the development of the TransMilenio Bus Rapid System and the restriction of vehicles through a daily, rotating ban on private cars depending on plate numbers.

However, this political restriction is less confining than it may first appear in that the Marxist historical framework is surprisingly flexible, and a rather simple matter to modify an alternative historical theory to use language that at least does not challenge the Marxist interpretation of history.

However, before this, some MPPC members got around this restriction by occasionally making longer stories in separate parts, and releasing them in successive weeks, starting with Vitagraph's The Life of Moses in five parts ( and five reels ) at the end 1909.

However, courts and various levels of government have indicated that the right is not without limit, and restrictions such as a prohibition of certain persons from owning or handling a gun ( convicted felons, mentally ill persons, persons under a restraining order ), a ban on certain types of ammunition ( armor-piercing, incendiary / tracer, explosive ), registration and restriction of certain types of firearms ( automatic weapons, short-barrelled shotguns, certain types of modifications ), and licensing requirements for possession or for public carry, have often passed judicial scrutiny ( although some of these points remain untested and others are currently under challenge ).

However, no randomized, controlled trial of oxalate restriction has yet been performed to test the hypothesis that oxalate restriction reduces the incidence of stone formation.

However, anyone may grow and use peyote, or Lophophora williamsii, as well as San Pedro and San Peruvianus without restriction, as it is specifically exempt from legislation.

However, CORBA does not place a restriction on the complexity of the calls.

However, the band was actually pleased with this nighttime viewing restriction.

However then Prime Minister, Harold Wilson rejected the proposal, feeling that it would appear to be an unacceptable restriction of the freedom of the public.

However, the proposal was not accepted, as there was no legal base for it and the then valid laws prohibited any restriction of pasture.

However, there is a height restriction of 4. 75 metres.

However, many modern schools of syntax – especially those that have been influenced by X-bar theory – make no such restriction.

However, according to a review in 2009, there is apparently no evidence of benefit of fluid restriction in patients with clinically stable heart failure otherwise receiving optimal pharmacological treatment.

However, the WTO, of which both the US and France are members, disallowed this restriction.

However, passes are available to visit the two museums on post, and the lake area remains open to the public without restriction since it is outside the cantonments.

However, there is no restriction for pepper gas pistol cartridges.

However, the last chapter of the book addresses limitations, especially for use in probability theory, of the restriction to locally compact spaces.

However, although a diphthong, such as oi, consists of two moras, stress may fall only on the first, a restriction not found with other vowel sequences such as io.

However, internal and external unrest, which the Tsar believed stemmed from political liberalisation, led to a series of repressions and a return to a former government of restriction and conservatism.

However, a political restriction on the numbers of people entering into the competitive market for services of the guild has the effect of raising the return on investments in the guilds training, especially for those already practising, by creating an artificial scarcity of guild members.

However, if the original conveyance was " to John Smith and his heirs for as long as John Smith does not use the premises to sell liquor, but if he does, then to the Red Cross " the rule would not be violated, because it would necessarily vest or expire within the life of John Smith ( since the restriction was on him alone ).

However, there are many other families of phosphodiesterases, including phospholipases C and D, autotaxin, sphingomyelin phosphodiesterase, DNases, RNases, and restriction endonucleases ( which all break the phosphodiester backbone of DNA or RNA ), as well as numerous less-well-characterized small-molecule phosphodiesterases.

( However, he started out as fully robotic, and was repaired with human parts, rather than the more usual reverse ).

However, it is more usual to use a linear transformation to transform these absolute values into values which resemble the more familiar Pauling values.

However, in the last two decades non-traditional El Niños were observed, in which the usual place of the temperature anomaly ( Nino 1 and 2 ) is not affected, but an anomaly arises in the central Pacific ( Nino 3. 4 ).

However, the usual way to show the link with the House of Orange-Nassau is the orange pennant above the red-white-blue.

However, grindcore alters the usual practices of metal or rock music in regard to song structure and tone.

However, there is a body of case law governing the civil commitment of individuals under the Fourteenth Amendment through U. S. Supreme Court rulings beginning with Addington v. Texas in 1979 which set the bar for involuntary commitment for treatment by raising the burden of proof required to commit persons from the usual civil burden of proof of " preponderance of the evidence " to the higher standard of " clear and convincing " evidence.

However, the red kryptonite and resulting powerlessness turned out to be a magical illusion created by Mister Mxyzptlk, with Superman's powers being restored once he learned of Mxyzptlk's involvement in accordance with the imp's usual restrictions.

However, there is also research suggesting that people are often helped by extended families and supportive religious leaders who listen with kindness and respect, which can often contrast with usual practice in psychiatric diagnosis and medication.

However, unlike the usual practice where the new year begins at midnight, the National New Year begins at the time determined by the astrologers.

However, in Byzantine and in Western European history, dynastic marriages were quite usual and there are many examples which were much more strange.

However, he prevented Louis XIV from sending the usual embassy of obedience to Alexander VII, and, while he lived, he foiled the appointment of a French ambassador to Rome, diplomatic affairs being meantime conducted by cardinal protectors, generally personal enemies of the Pope.

However, retroviruses function differently-their RNA is reverse-transcribed into DNA, which is integrated into the host cell's genome ( when it becomes a provirus ), and then undergoes the usual transcription and translational processes to express the genes carried by the virus.

However, some authors ( including mathematicians ) use φ for inclination ( or elevation ) and θ for azimuth, which " provides a logical extension of the usual polar coordinates notation ".

However, as usual, it is considered a best practice of VBScript style to be consistent and to capitalize judiciously.

However, the Gulf Arab States — Arafat's usual source for financial backing — still refused to provide him and the PLO with any major donations because of his sympathy for Iraq during the Gulf War, in 1991.

However, in 1822 he was proposed without his knowledge, and on election paid the usual fee.

However, unlike the ripieno concerto, which uses the usual ritornello form of the concerto, at least the first movement of these symphonies is in binary form.

However, the state of New York inverts the usual order, with the Supreme Court of the State of New York being the lowest trial court of general jurisdiction, and the Court of Appeals being the highest court.

After all, nothing in the usual definition said that the identity element e was unique ; if there is another identity element e, then it is ambiguous which one should be the value of the nullary operator e. However, this is not a problem because identity elements can be proved to be always unique.

However, Raeder was, as usual, not taking the long view.

However, in the usual use of the term, an MBO is a situation in which the management team initiates and actively pushes the acquisition.

However, there are remarkable sights of the more usual sort:

However, Sundin scored at a pace below his usual pace and retired in the off-season.

However, rather than the usual trapezoidal shape of a gaff sail, it is triangular, like a Bermuda rig.

However, Sigmund Feist ( 1909 ) rejects the theory on etymological grounds, as does Albert Morley Sturtevant ( 1951 ) on the grounds of major difficulties, and their points have led Bruce Lincoln ( 1977 ) to comment that " there is no reason whatever to contend that nagl-does not have its usual meaning of " nail " and that Naglfar is anything other than the nail-ship, just as Snorri describes it.