* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

For ƒ ∈ C ( X ) ( with a compact Hausdorff space X ), one sees that:

For instance, any continuous function defined on a compact space into an ordered set ( with the order topology ) such as the real line is bounded.

* For every natural number, the-sphere is compact.

For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of C as spectrum.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

For example, one 640MB compact disc ( CD ) holds approximately one hour of uncompressed high fidelity music, less than 2 hours of music compressed losslessly, or 7 hours of music compressed in the MP3 format at a medium bit rate.

For general nonabelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations.

For example, a compact Hausdorff space is metrizable if and only if it is second-countable.

For compact space | compact 2-dimensional surfaces without boundary ( topology ) | boundary, if every loop can be continuously tightened to a point, then the surface is topologically Homeomorphism | homeomorphic to a 2-sphere ( usually just called a sphere ).

For locally compact spaces an integration theory is then recovered.

* For a compact Hausdorff space X, the following are equivalent:

For example one can reconstruct $ X $ from C ( X ) when X is ( real ) compact.

For the space-optimized presentation of prefix tree, see compact prefix tree.

For compact enough objects, this solution generated a black hole with a central singularity.

For most Tandy 1000 models other than the compact EX and HX that did not come already equipped with a hard drive, Tandy offered hard disk options in the form of " hardcards " that were installed in one of the computer's expansion slots and consisted of a controller and drive ( typically a 3. 5 " MFM or RLL unit with a Western Digital controller ) mounted together on a metal bracket.

For Rousseau the remedy was not in going back to the primitive but in reorganizing society on the basis of a properly drawn up social compact, so as to " draw from the very evil from which we suffer civilization and progress the remedy which shall cure it.

For example, the product of the unit circle ( with its usual topology ) and the real line with the discrete topology is a locally compact group with the product topology and Haar measure on this group is not inner regular for the closed subset

For instance, the small calcite crystals in the sedimentary rock limestone and chalk change into larger crystals in the metamorphic rock marble, or in metamorphosed sandstone, recrystallization of the original quartz sand grains results in very compact quartzite, also known as metaquartzite, in which the often larger quartz crystals are interlocked.

For the VHS SP mode, which already uses a lower tape speed than the compact cassette, this resulted in a mediocre frequency response of roughly 100 Hz to 10 kHz for NTSC ; frequency response for PAL VHS with its lower standard tape speed was somewhat worse.

For example, a camera with a 1 / 1. 8 " sensor has a 5. 0x field of view crop, and so a hypothetical 5-50mm zoom lens produces images that look similar ( again the differences mentioned above are important ) to those produced by a 35mm film camera with a 25 – 250mm lens, while being much more compact than such a lens for a 35mm camera since the imaging circle is much smaller.

Again we verify the universal property: For f: X → K with compact Hausdorff and an ultrafilter on we have an ultrafilter on.

For with compact Hausdorff and an ultrafilter on we have an ultrafilter on, the pushforward of.

( 3 ) For anionics, these micelles appear to be roughly spherical assemblages in which the hydrocarbon tails come together so that the polar groups ( the ionized ends ) face outward towards the aqueous continuous phase.

For example, groups regularly canoe down the Connecticut River.

`` For instance, Djakarta, Indonesia, has three groups of dancers interested in coming here.

For they considered the odd numbers as male and the even ones as female, equating the two groups with the Yang and Yin principles in Nature ; ;

For a number of years, Wesleyan has been drawing varied groups of political and business leaders into these informal discussions with members of the faculty and student body, attempting to explore and clarify aspects of their responsibility for public policy.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

For example, of the isomers of butene, the two methyl groups of ( Z )- but-2-ene ( aka cis-2-butene ) face the same side of the double bond, and in ( E )- but2-ene ( aka trans-2-butene ) the methyl groups face the opposite side.

For example, there are differences between African, East Asian and Indo-racial groups in how they metabolize alcohol.

For example, in organic chemistry chemists are sometimes concerned only with the functional groups of the molecule.

For many years, conservative parties fought to stop extension of voting rights to groups such as to non-Christians, non-whites and women.

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

For groups 1 to 4, the number consists of three digits and a letter of the alphabet.

; Security: For security reasons, it is desirable to limit who can see or change specific attributes or groups of attributes.

groups ( G, *) and ( H, ●), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by.

For example, Lancaster students, motivated to save scrip, ultimately rented individual pages of textbooks from the school library, and read them in groups around music stands to reduce textbook costs.

For example Major General Charles J. Dunlap, Jr. says that United States Air Force culture includes an egalitarianism bred from officers as warriors who work with small groups of enlisted airmen either as the service crew or onboard crew of their aircraft.

For instance, although an individualist according to Cultural Theory is aversive towards both principles and groups, individualism is not fulfilling if individual brilliance cannot be recognised by groups, or if individual brilliance cannot be made permanent in the form of principles.

For example, a homomorphism of topological groups is often required to be continuous.

For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses.