( Sequentially compact )

* Sequentially compact spaces are countably compact.

Sequentially generated surrogate keys create the illusion that events with a higher primary key value occurred after events with a lower primary key value.

Sequentially, the Japanese 11th Army moved toward Lingling, seizing it on 4 September 1944, and controlled Guilin on 10 November 1944.

Sequentially, meditation comes as a prelude to contemplation.

Sequentially, while walking along a path which skirts the eroded channel of the river-formed natural stone bridge, one can see a pair of Vishnu sculptures with Lakshmi seated at his feet in a reclining pose.

* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.

* Every compact metric space is separable.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

* Countably compact: Every countable open cover has a finite subcover.

* Limit point compact: Every infinite subset has an accumulation point.

Every compact metric space is complete, though complete spaces need not be compact.

Every entire function can be represented as a power series that converges uniformly on compact sets.

* Every compact metric space ( or metrizable space ) is separable.

* Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.

Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.

Every continuous function on a compact set is uniformly continuous.

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space.

* Every compact Hausdorff space of weight at most ( see Aleph number ) is the continuous image of ( this does not need the continuum hypothesis, but is less interesting in its absence ).

*( BCT2 ) Every locally compact Hausdorff space is a Baire space.

Every group has a presentation, and in fact many different presentations ; a presentation is often the most compact way of describing the structure of the group.

*( BCT2 ) Every locally compact Hausdorff space is a Baire space.

Every H * is very special in structure: it is pure-injective ( also called algebraically compact ), which says more or less that solving equations in H * is relatively straightforward.

The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself: an open ( or half-open ) interval of the real numbers is not compact.

However, an open disk is not compact, because a sequence of points can tend to the boundary without getting arbitrarily close to any point in the interior.

Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can tend to the missing point without tending to any point within the space.

* A metric space ( or more generally any first-countable uniform space ) is compact if and only if every sequence in the space has a convergent subsequence.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

In 1953, Hidehiko Yamabe obtained the final answer to Hilbert ’ s Fifth Problem: a connected locally compact group is a projective limit of a sequence of Lie groups, and if " has no small subgroups " ( a condition defined below ), then G is a Lie group.

When an orthonormal basis that the vector a is represented in terms of is rotated, a's matrix in the new basis is obtained through multiplying a by a rotation matrix R. This matrix multiplication is just a compact representation of a sequence of dot products.

The donor is less massive than the compact object, and can be on the main sequence, a degenerate dwarf ( white dwarf ), or an evolved star ( red giant ).

Through them I believe that both the textual sequence of Holy Scripture and also a compact account of secular letters may, with God ’ s grace, be revealed .”

Indeed, if the interval were sequentially compact in constructive analysis, then the classical IVT would follow from the first constructive version in the example ; one could find c as a cluster point of the infinite sequence ( c < sub > n </ sub >)< sub > n </ sub >.

The instrument developed from the lute at an early date, being more compact and cheaper to build, but the sequence of development and nomenclature in different regions is now hard to discover.

* Exhaustion by compact sets, in analysis, a sequence of compact sets that converges on a given set

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.

In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions f < sub > n </ sub > on either metric space or locally compact space is continuous.

Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.

If M is a compact orientable manifold equipped with a smooth metric g, and Ω < sup > k </ sup >( M ) is the sheaf of smooth differential forms of degree k on M, then the de Rham complex is the sequence of differential operators

* K ( X, Y ) is a closed subspace of B ( X, Y ): Let T < sub > n </ sub >, n ∈ N, be a sequence of compact operators from one Banach space to the other, and suppose that T < sub > n </ sub > converges to T with respect to the operator norm.

This property is interesting in the case of locally compact topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, separable metric factor groups of G. More precisely, for a sequence

A compact sandstone sequence developed, about 20 x 30 kilometres wide and up to 600 metres thick dating to the lower Cenomanian to Santonian stages.

Vital to the residue theory of logarithmic forms is the Gysin sequence, which is in some sense a generalization of the Residue Theorem for compact Riemann surfaces.

( 2 ) Any sequence of continuous positive definite functions on G converging to 1 uniformly on compact subsets, converges to 1 uniformly on G.