* 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 compact groups, the Peter – Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of 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.

For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold — that is, one can smoothly " flatten out " certain manifolds, but it might require distorting the space and affecting the curvature or volume.

For example, symplectic topology — a subbranch of differential topology — studies global properties of symplectic manifolds.

For instance, calculus ( in one variable ) generalizes to multivariable calculus, which generalizes to analysis on manifolds.

For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature.

: For a different notion of boundary related to manifolds, see that article.

For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the oriented double cover.

For two dimensional manifolds, the above formula shows that if f is a harmonic function, then the conformal scaling g e < sup > 2ƒ </ sup > g does not change the Ricci curvature.

For the most reasonable finite-dimensional spaces ( such as compact manifolds, finite simplicial complexes or CW complexes ), the sequence of Betti numbers is 0 from some points onwards ( Betti numbers vanish above the dimension of a space ), and they are all finite.

For example, for ( 1 + 1 )- dimensional bordisms ( 2-dimensional bordisms between 1-dimensional manifolds ), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit ( trace ) or unit ( scalars ), depending on grouping of boundary, and thus ( 1 + 1 )- dimension TQFTs correspond to Frobenius algebras.

For probability manifolds such an inner product is given by the Fisher information metric.

For general-affine manifolds with one has:

For non-compact oriented manifolds, one has to replace cohomology by cohomology with compact support.

For example, a subclass of the K3 manifolds is elliptically fibered, and F-theory on a K3 manifold is dual to heterotic string theory on a two-torus.

For general Riemannian manifolds one has to add the curvature of ambient space ; if is a manifold embedded in a Riemannian manifold () then the curvature tensor of with induced metric can be expressed using the second fundamental form and, the curvature tensor of:

For n = 3, the h-cobordism theorem for smooth manifolds has not been proved and, due to the Poincaré conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structures.

For compact manifolds of dimension greater than 4, there is a finite number of " smooth types ", i. e. equivalence classes of pairwise smoothly diffeomorphic smooth structures.

For 1994, horsepower was reduced to 175, mostly due to the installation of smaller-volume exhaust manifolds ; torque ratings remained the same.

* For Riemannian manifolds of constant negative sectional curvature, any Jacobi field is a linear combination of, and, where.

* For Riemannian manifolds of constant positive sectional curvature, any Jacobi field is a linear combination of,, and, where.

For many engines, there are aftermarket tubular exhaust manifolds known as headers in US English, as ' extractors ' in Australian English, and simply as " tubular manifolds " in UK English.

For compact manifolds

For example, sheaves were applied to transformation groups ; as an inspiration to homology theory in the form of Borel-Moore homology for locally compact spaces ; to representation theory in the Borel-Bott-Weil theorem ; as well as becoming standard in algebraic geometry and complex manifolds.

For manifolds of dimension at most 6, any piecewise linear ( PL ) structure can be smoothed in an essentially unique way, so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds.