Suppose that X is a topological space. X is a completely regular space iff, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F. In fancier terms, this condition says that x and F can be separated by a function. X is a Tychonoff space, or T3½ space, or Tπ space, or completely T3 space if and only i ...

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Tychonoff space, Tychonoff space - Definitions, Tychonoff space - Examples and counterexamples, Tychonoff space - Properties

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In metric spaces, continuity and uniformity are usually defined in terms of δ’s and ε’s specifying numeric values of closeness. Intuitions from metric spaces transfer to topological spaces by thinking of a∈O, where O is a neighborhood of x, as a substitute for |x−a|<δ. The δ-ε definition of continuity translates directly into the topological definition. Similarly, metric intuitions transfer to uniformity by thinking of a∈U[x] as a substitute for |x ...

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Uniform space, Uniform space - History, Uniform space - Definition, Uniform space - Entourage definition, Uniform space - Uniform cover definition, Uniform space - Pseudometrics definition, Uniform space - Intuition, Uniform space - Examples, Uniform space - Uniformly continuous functions, Uniform space - Topology of uniform spaces, Uniform space - Completeness

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There are many who object to the idea of colonizing space as being too expensive and a waste of time. There is nothing in space that we really need, they say, adding that moving beyond the solar system is totally impractical in any reasonable time scale. The pragmatic argument to 'live together on the earth we have' is a powerful one, suggesting that if even half the money of space exploration were spent for terrestrial betterment, there would be greater good for a grea ...

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Space colonization, Space colonization - Method, Space colonization - Materials, Space colonization - Energy, Space colonization - Transportation, Space colonization - Communication, Space colonization - Life support, Space colonization - Radiation protection, Space colonization - Self-replication, Space colonization - Population size, Space colonization - Location, Space colonization - Orbit, Space colonization - Asteroid, Space colonization - Terrestrial analogues, Space colonization - Mercury, Space colonization - Venus, Space colonization - Europa, Space colonization - Gas Giants, Space colonization - Space habitats, Space colonization - Spaceship, Space colonization - Justification, Space colonization - Advocacy, Space colonization - Objections

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A zero-width space &#8203; is usually not visible, but it may expand in passages that are fully justified. In HTML pages this space can be used as a potential line-break in long words as a replacement for the deprecated <wbr> tag. Since the zero-width space is not supported in all web browsers (most notably the Internet Explorer) the following HTML code can be used as a workaround for this purpose: ...

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Space character, Space character - Spaces and digital typography, Space character - Zero-width space

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Let R denote the field of real numbers. For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R sometimes called real coordinate space and denoted Rn. An element of Rn is written x = (x1, x2, …, xn) where each xi is a real number. The vector space operations on R ...

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Euclidean space, Euclidean space - Real coordinate space, Euclidean space - Euclidean structure, Euclidean space - Alternative definition, Euclidean space - Euclidean topology

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Building cities in space will require materials, energy, transportation, communications, life support, and radiation protection. Space colonization - Materials. Colonies on the Moon and Mars can use local materials, although the Moon is deficient in carbon and nitrogen. For orbital colonies, launching materials from Earth is very expensive, so bulk materials should come from the Moon or Near-Earth Objects (NEOs - asteroids and comets with orbits near Earth) where gravitational forces are much less, there i ...

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Space colonization, Space colonization - Method, Space colonization - Materials, Space colonization - Energy, Space colonization - Transportation, Space colonization - Communication, Space colonization - Life support, Space colonization - Radiation protection, Space colonization - Self-replication, Space colonization - Population size, Space colonization - Location, Space colonization - Orbit, Space colonization - Asteroid, Space colonization - Terrestrial analogues, Space colonization - Mercury, Space colonization - Venus, Space colonization - Europa, Space colonization - Gas Giants, Space colonization - Space habitats, Space colonization - Spaceship, Space colonization - Justification, Space colonization - Advocacy, Space colonization - Objections

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Space Godzilla (スペースゴジラ - Supêsu Gojira, alternatively written SpaceGodzilla), is a modified clone of Godzilla featured in the VS Series of Godzilla films. In Godzilla vs. Space Godzilla (1994), the film which introduced the creature, characters hypothesize that Godzilla cells somehow cast into space fell into a black hole and reemerged, mutated and now a partially crystalline life form, from a white hole. As to how its cells had entered space in the first place, the film offers two possibilities: e ...

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Space Godzilla, Space Godzilla - Origins, Space Godzilla - Chronology, Space Godzilla - Powers, Space Godzilla - Notes, Space Godzilla - Conceptual origins

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Given any vector space V over some field F, we define the dual space V* to be the set of all linear functionals on V, i.e., scalar-valued linear transformations on V (in this context, a "scalar" is a member of the base-field F). V* itself becomes a vector space over F under the following definition of addition and scalar multiplication: for all φ, ψ in V*, a in F and x in V. In the language of tensors, elements of V are sometimes called covariant vectors, and elements of V*, contravariant vectors, covectors or one-forms.< ...

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Dual space, Dual space - Algebraic dual space, Dual space - Examples, Dual space - Transpose of a linear map, Dual space - Bilinear products and dual spaces, Dual space - Injection into the double-dual, Dual space - Continuous dual space, Dual space - Examples, Dual space - Further properties

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Uniform space - Entourage definition. A uniform space (X,Φ) is a set X equipped with a nonempty family of subsets of the Cartesian product X × X (Φ is called the uniform structure of X and its elements entourages (French:neighborhoods) or surroundings) with the following properties if U is in Φ, then U contains the diagonal { (x, x) : x in X }. if U is in Φ and V i ...

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Uniform space, Uniform space - History, Uniform space - Definition, Uniform space - Entourage definition, Uniform space - Uniform cover definition, Uniform space - Pseudometrics definition, Uniform space - Intuition, Uniform space - Examples, Uniform space - Uniformly continuous functions, Uniform space - Topology of uniform spaces, Uniform space - Completeness

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Space Wolves - Leman Russ. The Space Wolves Primarch Leman Russ, like all twenty Primarchs, mysteriously vanished from the Emperor's laboratories in his infancy. He turned up on the distant planet of Fenris, a snow-covered world inhabited by fairly primitive, warlike tribes of humans. He was raised by a pack of the massive Fenrisian Wolves that prowl the snowy lands. Among them, he grew ...

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Space Wolves, Space Wolves - History, Space Wolves - Leman Russ, Space Wolves - The Great Crusade, Space Wolves - After the Heresy, Space Wolves - Organization, Space Wolves - The Great Companies, Space Wolves - The Great Wolf, Space Wolves - Progression of a Space Wolf, Space Wolves - The Thirteenth Company, Space Wolves - Appearance, Space Wolves - Notable Members, Space Wolves - Background Inspiration

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Prior to the Horus Heresy, the Space Wolves Legion wore grey power armour, with a red snarling wolf as the Legion symbol. In the 41st millennium, Space Wolves power amour is a blue-grey, with other colours as highlights, most commonly red and yellow. The amour is often adorned with tokens taken from wolves, such as furs, tails and teeth. Great Company symbols vary, but are taken from Fenrisian mythology and ...

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Space Wolves, Space Wolves - History, Space Wolves - Leman Russ, Space Wolves - The Great Crusade, Space Wolves - After the Heresy, Space Wolves - Organization, Space Wolves - The Great Companies, Space Wolves - The Great Wolf, Space Wolves - Progression of a Space Wolf, Space Wolves - The Thirteenth Company, Space Wolves - Appearance, Space Wolves - Notable Members, Space Wolves - Background Inspiration

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Location is a frequent point of contention between space colonization advocates. The location of colonization can be: On a planet, natural satellite, or asteroid In orbit around the Earth, Sun, Lagrangian point or other object Main article: Colonization of Mars Mars is a frequent topic of discussion. Its overall surface area is similar to the dry land surface Earth, it has large water reserves, and has carbon (locked as carbon dioxide in the atmosphere). It may have gone throug ...

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Space colonization, Space colonization - Method, Space colonization - Materials, Space colonization - Energy, Space colonization - Transportation, Space colonization - Communication, Space colonization - Life support, Space colonization - Radiation protection, Space colonization - Self-replication, Space colonization - Population size, Space colonization - Location, Space colonization - Orbit, Space colonization - Asteroid, Space colonization - Terrestrial analogues, Space colonization - Mercury, Space colonization - Venus, Space colonization - Europa, Space colonization - Gas Giants, Space colonization - Space habitats, Space colonization - Spaceship, Space colonization - Justification, Space colonization - Advocacy, Space colonization - Objections

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Location is a frequent point of contention between space colonization advocates. The location of colonization can be: On a planet, natural satellite, or asteroid In orbit around the Earth, Sun, Lagrangian point or other object Main article: Colonization of Mars Mars is a frequent topic of discussion. Its overall surface area is similar to the dry land surface Earth, it may have large water reserves, and has carbon (locked as carbon dioxide in the atmosphere). It may have gone t ...

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Space colonization, Space colonization - Method, Space colonization - Materials, Space colonization - Energy, Space colonization - Transportation, Space colonization - Communication, Space colonization - Life support, Space colonization - Radiation protection, Space colonization - Self-replication, Space colonization - Population size, Space colonization - Location, Space colonization - Orbit, Space colonization - Asteroid, Space colonization - Terrestrial analogues, Space colonization - Mercury, Space colonization - Venus, Space colonization - Europa, Space colonization - Gas Giants, Space colonization - Space habitats, Space colonization - Spaceship, Space colonization - Justification, Space colonization - Advocacy, Space colonization - Objections

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One key feature of Space Invaders was the fact that as more and more of the aliens were shot, the remaining aliens would move faster and faster. The change in speed was minor at the beginning of a wave, but dramatic near the end. This action was originally an unintentional result of the way the game was written - as the program had to move fewer and fewer aliens, it could update the display faster - but the development team decided to retain this feature rather than implementing busy wa ...

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Space Invaders, Space Invaders - Description, Space Invaders - Implementation, Space Invaders - Graphics Design, Space Invaders - Legacy, Space Invaders - Popular response, Space Invaders - Later releases, Space Invaders - Street art, Space Invaders - Songs Inspired by Space Invaders

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One of the nicest properties of Hausdorff spaces is that limits of sequences, nets, and filters are unique whenever they exist. In fact, a topological space is Hausdorff if and only if every net (or filter) has at most one limit. Similarly, a space is preregular if all of the limits of a given net (or filter) are topologically indistinguishable. A useful alternative characterization of Hausdorff spaces is the following. A topological space X is Hausdorff iff the diagonal Δ = {(x,x) | x ∈ X} is closed as a sub ...

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Hausdorff space, Hausdorff space - Definitions, Hausdorff space - Examples and counterexamples, Hausdorff space - Properties, Hausdorff space - Preregularity versus regularity, Hausdorff space - Variants, Hausdorff space - Joke

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An algebraic space X is comprised of an affine scheme U and a closed subscheme R ⊂ U × U satisfying the following two conditions: 1. R is an equivalence relation as a subset of U × U 2. The projections pi: R → U onto each factor are étale maps. If a third condition 3. R is the trivial equivalence relation ov ...

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Algebraic space, Algebraic space - Definition, Algebraic space - Facts about algebraic spaces, Algebraic space - Applications, Algebraic space - Reference

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The game itself was an adaptation of the popular shooting gallery games that were a mainstay of carnivals. In this electronic version of the game, the player controlled the motions of a movable laser cannon that moved back and forth across the bottom of the video screen. Rows and rows of video aliens marched back and forth across the screen, slowly advancing down from the top to the bottom of the screen. If any of the aliens successfully landed on the bottom of the screen, the game would end. The player's laser cannon had an unlimited supply of ammunition to shoot at the aliens and destroy ...

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Space Invaders, Space Invaders - Description, Space Invaders - Implementation, Space Invaders - Graphics Design, Space Invaders - Legacy, Space Invaders - Popular response, Space Invaders - Later releases, Space Invaders - Street art, Space Invaders - Songs Inspired by Space Invaders

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For any metric space M, one can construct a complete metric space M' (which is also denoted as M with a bar over it), which contains M as a dense subspace. It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f' from M' to N which extends f. The space M' is determined up to isometry by this property, and ...

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Complete space, Complete space - Examples, Complete space - Some theorems, Complete space - Completion, Complete space - Topologically complete spaces, Complete space - Generalisations

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The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero. For instance, with the Lp spaces, the function defined by is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. These f ...

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Normed vector space, Normed vector space - Definition, Normed vector space - Topological structure, Normed vector space - Linear maps and dual spaces, Normed vector space - Normed spaces as quotient spaces of semi normed spaces, Normed vector space - Finite product spaces

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A discrete structure is often used as the "default structure" on a set that doesn't carry any other natural topology, uniformity, or metric. For example, any group can be considered as a topological group by giving it the discrete topology, implying that theorems about topological groups apply to all groups. Indeed, analysts may refer to the ordinary, non-topological groups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for ex ...

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Discrete space, Discrete space - Definitions, Discrete space - Properties, Discrete space - Uses, Discrete space - Indiscrete spaces, Discrete space - Quotation, Discrete space - Notes

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An important property of any Hilbert space is its reflexivity. In fact, more is true: one has a complete and convenient description of its dual space (the space of all continuous linear functions from the space H into the base field), which is itself a Hilbert space. Indeed, the Riesz representation theorem states that to every element φ of the dual H' there exists one and only one uSee also:

Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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Two bullfrogs were launched on a one way mission on the Orbiting Frog Otolith satellite on November 9, 1970 to better understand space motion sickness. Apollo 16 on April 16, 1972 carried nematodes and Apollo 17, launched on December 7, 1972 carried five pocket mice, although one died on the circumlunar trip. Skylab 3 carried pocket mice and the first fish in space (mummichog), and the first spiders in space, cross spiders. The U.S. also f ...

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Animals in space, Animals in space - 1940s, Animals in space - 1950s, Animals in space - 1960s, Animals in space - 1970s, Animals in space - 1980s, Animals in space - 1990s, Animals in space - 2000s, Animals in space - External link

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