The most general stationary black hole solution known is the Kerr – Newman metric, which describes a black hole with both charge and angular momentum.

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces.

Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces.

While all these conditions are equivalent for metric spaces, in general we have the following implications:

In addition, this article discusses the definition for the more general case of functions between two metric spaces.

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.

A Finsler metric is a much more general structure than a Riemannian metric.

This expression is correct in the full theory of general relativity, to lowest order in the gravitational field, and ignoring the variation of the space-space and space-time components of the metric tensor, which only affect fast moving objects.

The effect can also be derived by using either the exact Schwarzschild metric ( describing spacetime around a spherical mass ) or the much more general post-Newtonian formalism.

Einstein ’ s general theory modifies the distinction between nominally " inertial " and " noninertial " effects by replacing special relativity's " flat " Euclidean geometry with a curved metric.

Nets are one of the many tools used in topology to generalize certain concepts that may only be general enough in the context of metric spaces.

However in general the Ricci flow equations lead to singularities of the metric after a finite time.

* de Sitter precession a general relativistic correction accounting for the Schwarzschild metric of curved space near a large non-rotating mass.

* Lense-Thirring precession a general relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.

Various ideas from real analysis can be generalized from real space to general metric spaces, as well as to measure spaces, Banach spaces, and Hilbert spaces.

However, it is a long-standing ( since 1975 ) open problem to improve the Christofides approximation factor of 1. 5 for general metric TSP to a smaller constant.

Like the general TSP, the Euclidean TSP ( and therefore the general metric TSP ) is NP-complete.

However, in some respects it seems to be easier than the general metric TSP.

In non-Cartesian ( non-square ) or curved coordinate systems, the Pythagorean theorem holds only on infinitesimal length scales and must be augmented with a more general metric tensor g < sub > μν </ sub >, which can vary from place to place and which describes the local geometry in the particular coordinate system.

Physicists have long been aware that there are solutions to the theory of general relativity which contain closed timelike curves, or CTCs — see for example the Gödel metric.

Using the ADM formalism of general relativity, the spacetime is described by a foliation of space-like hypersurfaces of constant coordinate time t. The general form of the metric described within the context of this formalism is:

In general, such apartments afford more protection than smaller buildings because their walls are thick and there is more space.

This he did by using utterly literal means to carry the forward push of the collage ( and of Cubism in general ) literally into the literal space in front of the picture plane.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

He introduced the concept of a uniform space in general topology, as a by-product of his collaboration with Nicolas Bourbaki ( of which he was a Founding Father ).

Math., 1858 ) and Ernst Abbe showed that the properties of these reproductions, i. e. the relative position and magnitude of the images, are not special properties of optical systems, but necessary consequences of the supposition ( in Abbe ) of the reproduction of all points of a space in image points ( Maxwell assumes a less general hypothesis ), and are independent of the manner in which the reproduction is effected.

In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball.

The framework for the Big Bang model relies on Albert Einstein's general relativity and on simplifying assumptions such as homogeneity and isotropy of space.

In a more general notation, for any basis in 3d space we write ;

The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was not fully appreciated for another four decades.

It is one of five arguments from the " properties, causes, and effects " of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space.

In 1916, Albert Einstein published his theory of general relativity, which provided a unified description of gravity as a geometric property of space and time.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness — originally called bicompactness — involves families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

This ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space.

It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.

Free from many of the elements that accompany science fiction in general — whether that be space aliens, giant robots, or laser guns — the series delegates itself towards presenting a world that is quite similar to our own albeit showcasing some technological advances.

In it, he puts forth a new theory about the nature of space and describes how this theory influences thinking about architecture, building, planning, and the way in which we view the world in general.