In mathematics , a pseudometricspace is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space , every seminormed space is a pseudometricspace. Because of this analogy the term semimetric space which has a different ... WP MATH for why. Unlike a metric space, points in a pseudometricspace need not be identity of indiscernibles ... in X math . This point then induces a pseudometric on the space of functions, given by math , d f,g f x 0 g x 0 math for math f,g in mathcal F X math For vector spaces V , a seminorm p induces a pseudometric on V , as math , d x,y p x y . math Conversely, a homogeneous, translation invariant pseudometric induces a seminorm. Topology The pseudometric topology is the topological space topology induced .... ref planetmath reference id 6284 title Pseudometric topology ref A topological space is said to be a pseudometrizable topological space if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space. The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is T0 space T sub 0 sub i.e. distinct points are topologically distinguishable ... the metric identification , that converts the pseudometricspace into a full fledged metric space . This is done ... planetmath id 6273 title Pseudometricspace planetmath reference id 6275 title Example of pseudometricspace DEFAULTSORT PseudometricSpace Category Properties of topological spaces Category Metric ... a topology is generated using a family of pseudometrics, the space is called a gauge space . Definition A pseudometricspace math X,d math is a set math X math together with a non negative real valued function math d X times X longrightarrow mathbb R geq 0 math called a pseudometric such that, for every ... . Examples Pseudometrics arise naturally in functional analysis . Consider the space math mathcal ... more details
The Space may refer to The Space Connecticut , a music and arts venue The Space Theatre in London See also Special Prefixindex The Space The Space all titles starting from Special Prefixindex The space The space all titles starting from Space disambiguation disambig ... more details
The term SPACE capitalized can refer to Space TV channel SPACE , a Canadian science fiction channel The Society for Promotion of Alternative Computing and Employment DSPACE , a term in computational complexity theory Spectroscopic All Sky Cosmic Explorer , a proposed satellite designed to measure the baryon acoustic oscillations disambig ... more details
pp vandalism small yes pp move indef About the general framework of distance and direction the space beyond Earth s atmosphere Outer space all other uses Space disambiguation Space is the boundless, three .... ref http www.britannica.com eb article 9068962 space Britannica Online Encyclopedia Space ref Physical space is often conceived in three linear dimension s, although modern physics physicists ... and with different underlying structures. The concept of space is considered to be of fundamental importance ... framework . Debates concerning the nature, essence and the mode of existence of space date back to antiquity ... the Greeks called khora i.e. space , or in the Physics of Aristotle Book IV, Delta in the definition of topos i.e. place , or even in the later geometrical conception of place as space qua extension in the Discourse ... . In Isaac Newton s view, space was absolute in the sense that it existed permanently and independently of whether there were any matter in the space. ref French and Ebison, Classical Mechanics, p. 1 ref Other natural philosopher s, notably Gottfried Leibniz , thought instead that space was a collection ... ian Immanuel Kant said neither space nor time can be empirically perceived, they are elements of a systematic framework that humans use to structure all experiences. Kant referred to space in his ... non Euclidean geometries , in which space can be said to be curved , rather than flat . According to Albert Einstein s theory of general relativity , space around gravitational field s deviates from Euclidean space. ref Carnap, R. An introduction to the Philosophy of Science ref Experimental tests of general relativity have confirmed that non Euclidean space provides a better model for the shape of space. Philosophy of space Leibniz and Newton Image Gottfried Wilhelm von Leibniz.jpg Gottfried Leibniz right thumb 150px In the seventeenth century, the philosophy of space and time emerged ... theories of what space is. Rather than being an entity that independently exists over and above other ... more details
Unreferenced auto yes date December 2009 Orphan date December 2009 Infobox television show name This Space format Sitcom runtime 30 minutes creator Jason Howard starring Jason Howard br country United Kingdom network ITV first aired August 18, 2006 This Space is a situation comedy from United Kingdom. There were 6 episodes. Category British television sitcoms UK tv prog stub ... more details
lowercase t Space Citations missing date July 2008 Infobox company name Transformational Space Corporation br t Space logo Image TSpace logo.jpg 150px type Private company Private genre Commercial spaceflight ... www.transformspace.com footnotes intl t Space or Transformational Space Corporation is an United ... to the International Space Station . The company is headquartered in Reston, Virginia . History In September 2004 t Space was one of eleven companies selected by NASA to conduct preliminary concept ... t Space Wins NASA Lunar Exploration Contract work SpaceRef.com date 2004 09 02 accessdate 2008 07 04 ... , and Northrop Grumman to provide vehicle and architecture advice to NASA for the Vision for Space ... http www.msnbc.msn.com id 11927039 page 2 title Private ventures vie to service space station last ... 2007, NASA signed a Space Act Agreement with t Space on technical assistance and data access ... and to private space habitats. Citation needed date October 2009 In 2011, t Space proposed an eight ... that could be sent into space on a variety of launch vehicles. ... Up to eight crew, Soyuz like architecture ... seat basis. Nominal land recovery with water backup. ref The t Space proposal was not selected for NASA ... Space Shuttle url http www.aviationweek.com aw generic story generic.jsp?channel awst&id news ..., ... went to Blue Origin, Boeing, Sierra Nevada Corp. and Space Exploration Technologies Inc. SpaceX ... Two to the ISS, is Vice President of Space Exploration Technology and Bretton Alexander, a former ... and External Affairs. cn CXV proposal t Space was working on designs when for an air launch air launched passenger carrying space capsule capsule termed the Crew Transfer Vehicle, or CXV. In contrast to the Space Shuttle and other companies CEV proposals, this craft would be specialized .... cn Under their plan, the CXV would have been capable of docking with the International Space Station , a Bigelow Commercial Space Station commercial habitat such as those being developed by Bigelow ... more details
Infobox Album See Wikipedia WikiProject Albums Name In Space Type Album Artist Big Star band Big Star Cover Inspace.jpg Released 27 September 2005 Recorded 2004 Genre Power pop Length 39 04 Label Rykodisc Producer Last album Columbia Live at Missouri University Columbia Live at Missouri University 4 25 93 br 1993 This album In Space br 2005 Next album Keep an Eye on the Sky br 2009 Album ratings rev1 Allmusic rev1score Rating 3 5 ref Allmusic class album id r794064 pure url yes Allmusic review ref Automatically generated by DASHBot In Space is the fourth studio album by the United States American rock group Big Star band Big Star , released in 2005. It was the first new studio recording by the band since Third Sister Lovers , which was recorded in 1974. The album featured original Big Star members Alex Chilton and Jody Stephens , with two more recent recruits, Jon Auer and Ken Stringfellow . Auer and Stringfellow had been members of The Posies before joining Big Star for sporadic live performances and tours since 1993. ref Big Star at Allmusic Allmusic class artist id p3676 pure url yes ref Big Star did not tour in support of the album. Mine Exclusively was originally recorded by The Olympics band The Olympics in 1966. Track listing All songs were written by Auer, Chilton, Stephens, and Stringfellow, except where noted. Dony 2 45 Lady Sweet 3 40 Best Chance 3 04 Turn My Back on the Sun 2 38 Love Revolution 5 49 Auer, Chilton, Jim Spake, Stephens, Stringfellow, Nokie Taylor February s Quiet 2 44 Mine Exclusively 2 31 Shirley Mae Matthews A Whole New Thing 3 52 Aria, Largo 2 30 Georg Muffat Hung Up with Summer 3 03 Auer, Chilton, Bill Cunningham musician Bill Cunningham , Stephens, Stringfellow Do You Wanna Make It 2 46 Makeover 3 42 Personnel Jon Auer guitar , vocals Alex Chilton guitar, vocals Jody Stephens drum kit drums , vocals Ken Stringfellow bass guitar , vocals References ... more details
for the community center in Philadelphia A Space community center File Aspace logo.png thumb A Space Logo The United States Intelligence Community A Space , or Analytic Space , is a project from the Office of the Director of National Intelligence s ODNI Office of Analytic Transformation and Technology ... phase of A Space. Initial operational capability was scheduled for December 2007. ref name IW2007 ...?articleID 201801990 date 23 August 2007 ref A Space went live on the government s classified Joint Worldwide Intelligence Communications System Sept. 22, 2008. ref citation title A Space set to launch ... 153673 1.html date September 3, 2008 ref A Space is built on Jive Software s Clearspace application ... 2009 07 23 Social networking media national security.aspx?Page 2 date 23 July 2009 ref A Space is part ... A Space, according to Andrew McAfee of Harvard Business School , ref name McAfee2007 citation url ... or A Space encourages collaboration among people with weak ties, complementing the traditional team ... desktop client. Information Week stated that A Space must have 16 different security waivers and move ..., the users themselves need to be monitored. One of the ways A Space will maintain its security will be through ... large databases of current intelligence and Intellipedia . Eventually, A Space will be capable of carrying ... of the common interests. blockquote Social networking will be another critical part of A Space ... A Space is open to all elements of the Intelligence Community. One of the first DIA components to enter A space will be a new Counter intelligence Counterintelligence force protection source .... DNI policy is to have A Space rely, as much as possible, on commercial off the shelf COTS software ... technical mode for building A space. Eventually, that will allow analysts and developers to write ..., and A Space could therefore take advantage of RSS feeds for blog content, among other things. It will be implemented ... Intelligence Agency Category Intelligence analysis es A Space it US Intelligence Community A Space scn ... more details
In mathematics , a topological space X is uniformizable if there exists a uniform structure on X which Uniform space Topology of uniform spaces induces the topology of X . Equivalently, X is uniformizable if and only if it is homeomorphic to a uniform space equipped with the topology induced by the uniform structure . Any pseudometricspace pseudo metrizable space is uniformizable since the pseudo metric uniformity induces the pseudo metric topology. The converse fails There are uniformizable spaces which are not pseudo metrizable. However, it is true that the topology of a uniformizable space can always be induced by a family of Pseudometricspacepseudometric s indeed, this is because any uniformity on a set X can be uniform space Pseudometrics definition defined by a family of pseudometrics. Showing that a space is uniformizable is much simpler than showing it is metrizable. In fact, uniformizability is equivalent to a common separation axiom A topological space is uniformizable if and only if it is completely regular . Induced uniformity One way to construct a uniform structure on a topological space X is to take the initial uniformity on X induced by C X , the family of real valued continuous function s on X . This is the coarsest uniformity on X for which all such functions ... space X there is a finest uniformity on X compatible with the topology of X called the fine uniformity or universal uniformity . A uniform space is said to be fine if it has the fine uniformity ... continuous function f from a fine space X to a uniform space Y is uniformly continuous. This implies that the functor F CReg Uni which assigns to any completely regular space X the fine uniformity on X is left adjoint to the forgetful functor which sends a uniform space to its underlying completely regular space. Explicitly, the fine uniformity on a completely regular space X is generated by all ... D n circ D n subset D n 1 math . The uniformity on a completely regular space X induced by C X ... more details
of pseudometricspace pseudometrics , an approach which is particularly useful in functional analysis with pseudometrics provided by seminorm s . More precisely, let f X X R be a pseudometric on a set ... , such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space ... space. Indeed, since a metric is a fortiori a pseudometric, the Uniform space Pseudometrics definition ...No footnotes date May 2009 In the mathematical field of topology , a uniform space is a Set mathematics set with a uniform structure . Uniform spaces are topological space s with additional structure which is used to define uniform property uniform properties such as complete space completeness , uniform ... structure s is that in a uniform space, one can formalize certain notions of relative closeness and closeness ... spaces. By comparison, in a general topological space, given sets A,B it is meaningful to say that a point ... by topological structure alone. Uniform spaces generalize metric space s and topological group s and therefore ... definitions for a uniform structure. Entourage definition A uniform space X , is a Set mathematics ... in If the last property is omitted we call the space quasiuniform . One usually writes U x y x ... a set of B . Every uniform space has a fundamental system of entourages consisting of symmetric entourages. The right intuition about uniformities is provided by the example of metric space s if X , d is a metric space, the sets math U a x,y in X times X d x,y leq a quad text where quad a 0 ... by the single pseudometric f . Certain authors call spaces the topology of which is defined in terms ... structure is defined by a single pseudometric, namely the upper envelope sup f sub i sub of the family ... can be defined by a single pseudometric. A consequence is that any uniform structure can be defined ... IX 1 no. 4 . Uniform cover definition A uniform space X , is a set X equipped with a distinguished ... x as a typical neighbourhood of x of size P , and this intuitive measure applies uniformly over the space ... more details
Separation axiom In topology and related branches of mathematics , a topological space X is a T sub 0 sub space or Kolmogorov space if for every pair of distinct points of X , at least one of them has ... . These spaces are named after Andrey Kolmogorov . Definition A T sub 0 sub space is a topological space in which every pair of distinct points is topologically distinguishable . That is, for any ... being separated. In a T sub 0 sub space, the second arrow above reverses points are distinct if and only ... are T sub 0 sub . In particular, all Hausdorff space Hausdorff T sub 2 sub spaces and T1 space T sub 1 sub space s are T sub 0 sub . Spaces which are not T sub 0 sub A set with more than one element .... The space of all measurable function s f from the real line R to the complex plane C such that the Lebesgue ... 1 sub since the particular point is not closed its closure is the whole space . An important special case is the Sierpi ski space which is the particular point topology on the set 0,1 . The excluded ... T sub 0 sub space is of this type. This also includes the particular point and excluded point topologies ... 0. Quite generally, a topological space X will be T sub 0 sub if and only if the specialization preorder ... i.e. agrees with equality . So a space will be T sub 0 sub but not T sub 1 sub if and only if the specialization ... of topological space typically studied are T sub 0 sub . Indeed, when mathematicians in many fields ..., consider a well known example. The space Lp space L sup 2 sup R is meant to be the space of all ... of f x sup 2 sup over the entire real line is finite set finite . This space should become a normed vector space by defining the norm f to be the square root of that integral. The problem is that this is not really ... space of the original seminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties from the seminormed space see below. In general, when dealing ... that can be placed on a topological space. The Kolmogorov quotient Topological indistinguishability ... more details
s are perfectly normal Hausdorff All pseudometricspace s and hence all pseudometrisable space s are perfectly ...for normal vector space normal geometry Separation axiom In topology and related branches of mathematics , a normal space is a topological space X that satisfies Axiom T sub 4 sub every two disjoint closed set s of X have disjoint open neighborhood s. A normal Hausdorff space Hausdorff space is also called a T sub 4 sub space . These conditions are examples of separation axiom s and their further strengthenings ... Hausdorff spaces , or T sub 6 sub spaces . Definitions A topological space X is a normal space if, given ... 4 sub space is a T1 space T sub 1 sub space X that is normal this is equivalent to X being Hausdorff space Hausdorff and normal. A completely normal space or a hereditarily normal space is a topological space X such that every subspace topology subspace of X with subspace topology is a normal space ... by neighbourhoods. A completely T sub 4 sub space , or T sub 5 sub space is a completely normal Hausdorff topological space X equivalently, every subspace of X must be a T sub 4 sub space. A perfectly normal space is a topological space X in which every two disjoint non empty closed sets E and F ... closed set is a zero set . Every perfectly normal space is automatically completely normal. Fact date August 2008 A Hausdorff perfectly normal space X is a T sub 6 sub space , or perfectly T sub 4 sub space . Note that the terms normal space and T sub 4 sub and derived concepts occasionally have ... . Terms like normal regular space and normal Hausdorff space also turn up in the literature &mdash they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T sub 4 sub space. Given the historical confusion ... space s and paracompact Hausdorff space fully T sub 4 sub space s are discussed elsewhere they are related to paracompactness . A locally normal space is a topological space where every point has ... more details
For the concept in set theory Baire space set theory In mathematics , a Baire space is a topological space which, intuitively speaking, is very large and has enough points for certain limit processes. It is named ... space, the class of closed set s with empty set empty interior topology interior consists precisely ... subspaces in a Euclidean space . A topological space is a Baire space if it is large , meaning that it is not a countable ... Euclidean space is not a countable union of its affine planes. Definition The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints ... which is closer to the definition originally given by Baire. Modern definition A topological space is called a Baire space if the union set theory union of any countable Class set theory collection of closed ... unrelated to category theory as follows. A subset of a topological space X is called nowhere dense ... for a Baire space can then be stated as follows a topological space X is a Baire space if every non .... A topological space X is a Baire space if and only if every comeager subset of X is dense. Examples The space R of real number s with the usual topology, is a Baire space, and so is of second category ... in R . The Cantor set is a Baire space, and so is of second category in itself, but it is of first ... enumerates the rational number s. Note that the space of rational number s with the usual topology inherited from the real number reals is not a Baire space, since it is the union of countably many closed ... category theorem The Baire category theorem gives sufficient condition s for a topological space to be a Baire space. It is an important tool in topology and functional analysis . BCT1 Every complete space complete metric space is a Baire space. More generally, every topological space which is homeomorphic to an open set open subset of a complete space complete pseudometricspace is a Baire space. In particular, every topologically complete space is a Baire space. BCT2 Every locally compact Hausdorff ... more details
spaces are kinds of topological space s. These conditions are examples of separation axiom s. Tychonoff ... that X is a topological space. X is a completely regular space if given any closed set F and any Point ... can be separated sets separated by a continuous function . X is a Tychonoff space , or T sub 3 sub space , or T sub sub space , or completely T sub 3 sub space if it is both completely regular and Hausdorff space Hausdorff . Note that some mathematical literature uses different definitions for the term ..., is unambiguous, and always means a Tychonoff space. For more on this issue, see History of the separation ... of Kolmogorov equivalence . A topological space is Tychonoff if and only if it s both completely regular and Kolmogorov space T sub 0 sub . On the other hand, a space is completely regular if and only if its Kolmogorov quotient is Tychonoff. Examples and counterexamples Almost every topological space ... line is Tychonoff under the standard Euclidean space Euclidean topology . Other examples include Every metric space is Tychonoff every pseudometricspace is completely regular. Every locally compact regular space regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff. In particular, every topological manifold is Tychonoff. Every totally ordered ... both the metric spaces and the topological groups, every uniform space is completely regular. The converse is also true every completely regular space is uniformisable. Every CW complex is Tychonoff. Every normal space normal regular space is completely regular, and every normal Hausdorff space is Tychonoff. The Niemytzki plane is an example of a Tychonoff space which is not normal space normal ... space has the same property. A nonempty product space is completely regular resp. Tychonoff if and only if each factor space is completely regular resp. Tychonoff . Like all separation axioms, completely ... space quotients of completely regular spaces need not be regular space regular . Quotients of Tychonoff ... more details
. In general this will only define a pseudometricspacepseudometric , i.e. math d x , y 0 math does ... of a pseudometricspace , a quasimetric space , or a semimetric space . If the distance function ...In mathematics , a metric space is a Set mathematics set where a notion of distance called a metric mathematics metric between elements of the set is defined. The metric space which most closely corresponds to our intuitive understanding of space is the 3 dimensional Euclidean space . In fact, the notion ... of the Straight line straight line segment connecting them. The geometric properties of the space ... geometries such as those used in the theory of general relativity . A metric space also induces ... abstract topological space s. History Expand section Reasons for generalizing the Euclidean metric, first ... 22 1906 1 74. Definition A metric space is an ordered pair math M,d math where math M math is a set ... writes math M math for a metric space if it is clear from the context what metric is used. Examples of metric spaces Finite Metric space redirects here Ignoring mathematical details, for any system of roads ... y x vert math given by the absolute difference , and more generally Euclidean space Euclidean math n math space with the Euclidean distance , are complete space complete metric spaces. The rational number s with the same distance also form a metric space, but are not complete. The positive real numbers with distance function math d x,y vert log y x vert math is a complete metric space. Any normed vector space is a metric space by defining math d x,y lVert y x rVert math , see also Metric 28mathematics 29 Relation of norms and metrics relation of norms and metrics . If such a space is Complete metric space complete , we call it a Banach space . Examples The Norm mathematics Taxicab norm or Manhattan ... also called the Post Office metric or the SNCF metric on a normed vector space is given by math d x ... space and math X math is a subset of math M math , then math X,d math becomes a metric space by restricting ... more details
converges to both 0 and 1. Metrizability The Sierpi ski space S is not metrizable space metrizable or even pseudometrizable space pseudometrizable since it is not regular. S is generated by the hemimetric or pseudometricspace pseudo quasimetric math d 0,1 0 math and math d 1,0 1 math . Other properties ...In mathematics , the Sierpi ski space or the connected two point set is a finite topological space with two ... space which is neither trivial topology trivial nor discrete topology discrete . It is named after Wac aw Sierpi ski . The Sierpi ski space has important relations to the Computability theory theory of computation ... Explicitly, the Sierpi ski space is a topological space S whose underlying set mathematics point set ... 1 0,1 . math A finite topological space is also uniquely determined by its specialization preorder . For the Sierpi ski space this preorder is actually a partial order and given by math 0 leq 0, qquad 0 leq 1, qquad 1 leq 1. math Topological properties The Sierpi ski space S is a special case ... which contains only one of these points. Therefore S is a Kolmogorov space Kolmogorov T sub 0 sub space . However, S is not T sub 1 sub since the point 1 is not closed. It follows that S is not Hausdorff space Hausdorff , or T sub n sub for any n &ge 1. S is not regular space regular or completely ... vacuously normal space normal and completely normal space completely normal since there are no nonempty separated set s. S is not perfectly normal space perfectly normal since the disjoint closed sets ... constant . Connectedness The Sierpi ski space S is both hyperconnected since every nonempty open ... connected space connected and path connected space path connected . A path topology path from ... path connected . The Sierpi ski space is contractible space contractible , so the fundamental ... topological spaces, the Sierpi ski space is both compact space compact and second countable . The compact ... normal space fully normal . ref Steen and Seebach incorrectly list the Sierpi ski space as not being ... more details
stated in their definitions. A simple example of a topology that is T1 space T sub 1 sub but is not Hausdorff is the cofinite topology defined on an infinite set . Pseudometricspace s typically are not Hausdorff ...Separation axiom In topology and related branches of mathematics , a Hausdorff space , separated space or T sub 2 sub space is a topological space in which distinct points have disjoint sets disjoint neighbourhood ... space, the Hausdorff condition T sub 2 sub is the most frequently used and discussed. It implies ... net s, and filter topology filter s. Intuitively, the condition is illustrated by the pun that a space ... space in 1914 included the Hausdorff condition as an axiom. Definitions Image Hausdorff space.svg ... x and y in a topological space X can be separated by neighbourhoods if existential quantification there exists ... are Disjoint sets disjoint nowrap U V &empty . X is a Hausdorff space if any two distinct points ... space is also used. A related, but weaker, notion is that of a preregular space . X is a preregular space if any two topologically distinguishable points can be separated by neighbourhoods. Preregular .... A topological space is Hausdorff if and only if it is both preregular i.e. topologically distinguishable points are separated and Kolmogorov space Kolmogorov i.e. distinct points are topologically distinguishable . A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff. Equivalences For a topological space X , the following are equivalent X is a Hausdorff space. Limits of net topology nets in X are unique. ref Willard, pp. 86&ndash 87. ref Limits of filter ... x . The diagonal x , x x X is closed set closed as a subset of the product space X × X . Examples ... space. More generally, all metric space s are Hausdorff. In fact, many spaces of use in analysis ... of Hausdorff gauge space s. Indeed, when analysts run across a non Hausdorff space, it is still ... complete Heyting algebra is the algebra of open set s of some topological space, but this space need ... more details
In topology , approach spaces are a generalization of metric space s, based on point to set distances, instead of point to point distances. They were introduced by http www.math.ua.ac.be TOP Robert Lowen in 1989. Definition Given a metric space X , d , or more generally, an Metric mathematics Extending the range extended Pseudometricspace pseudo quasimetric which will be abbreviated xpq metric here , one can define an induced map d X × P X 0, by d x, A inf d x , a a A . With this example in mind, a distance on X is defined to be a map X × P X 0, satisfying for all x in X and A , B X , d x , x 0 d x , d x , A B min d x , A , d x , B For all , 0 , d x , A d x , A sup sup where A sup sup x d x , A by definition. The empty infimum is positive infinity convention is like the Empty product Nullary intersection nullary intersection is everything convention. An approach space is defined to be a pair X , d where d is a distance function on X . Every approach space has a topology, given by treating A A sup 0 sup as a Kuratowski closure axioms Kuratowski closure operator . The appropriate maps between approach spaces are the contractions . A map f X , d Y , e is a contraction if e f x , f A d x , A for all x X , A X . Examples Every xpq metric space X , d can be distancized to X , d , as described at the beginning of the definition. Given a set X , the discrete distance is given by d x , A 0 if x A and if x A . The induced topology is the discrete topology. Given a set X , the indiscrete distance is given by d x , A 0 if A is non empty, and if A is empty. The induced topology is the indiscrete topology. Given a topological space X , a topological distance ... . Given any approach space X , d , the maps for each A X d ., A X , d ..., and let e , A if A is bounded. Then P , e is an approach space. Topologically, P is the one ... b U , A sup inf n j n X , j E X U , E F A . Then N , b is an approach space that extends the ordinary ... more details
Infobox Fictional Spacecraft name Space Dock image Spacedock.jpg caption Space Dock aka Interplanetary Space Station , Meta Probe Launch Platform & Centauri Space Station first Breakaway Space 1999 Breakaway ... Orbital fighters Mark IX Hawk s auxcraft Eagle Space 1999 Eagle Transporter s armaments defense propulsion The Space Dock was a fictional space station in the television series Space 1999 1975 . Referred to simply as the Space Dock it is also called the Meta Probe Launch Platform in Breakaway Space 1999 Breakaway and the Interplanetary Space Station in Dragon s Domain episodes of the series. The popular fan name for the space station, Centuri , was coined in The Moonbase Alpha Technical Manual ... of the Mark IX Hawk seen in the episode War Games Space 1999 Season 1 .281975 1976.29 War ..., there are proposed plans to build a space dock to function as a type of dry dock or spaceport making possible the on orbit assembly and maintenance of large space facilities, space platforms, and spacecraft ... in Space Logistics Capabilities updated.pdf Architecting Rapid Growth in Space Logistics Capabilities ... The Space Dock is used for launching interplanetary missions such as the Ultra Probe 1996 ref Space 1999 episode Dragon s Domain ref and the Meta Probe 1999 ref name Space 1999 Space 1999 episode Breakaway Space 1999 Breakaway ref , as well as providing a midway point for travellers to and from the Moon ... on 13 September 1999, the Space Dock was destroyed by extreme gravitational stress although the GTV News announcer reports that it, like the Moon, was hurled out of orbit. ref name Space 1999 History Under the guidance of the International Lunar Finance Commission, on 2 July 1981 the Space ... to oversee construction of Earth s Space Research Centre, or Moonbase Alpha , which began ... lengths projecting outward from the top and bottom of the core. These arms of the Space Dock appear to have been Saturn rocket s, such as those used in early space exploration. With their standardised ... more details
In mathematics , Moore space may refer to Moore space algebraic topology Moore space topology , a regular, developable topological space. mathdab ... more details
Can refer to Znamya space mirror Znamya , an orbital space mirror Space mirror anti global warming measure Space Mirror Memorial , an astronaut memorial at the Kennedy Space Center Disambig ... more details
Space Corps , may refer to Space Corps comics , the outer space arm of the Mega City One Judges Space Corps, the organisation in Red Dwarf, part of the running jokes involving Space Corps Directives disambig ... more details
The Space Based Space Surveillance SBSS system is a planned constellation of satellite s and supporting ground infrastructure that will improve the ability of the United States Department of Defense DoD to detect and track space objects in orbit around the Earth. The primary SBSS contractor, Boeing , characterizes some orbiting space objects as, Potential future threats to the United States space assets. ref name gimble cite web url http www.boeing.com defense space ic sis news 2008 q2 080421b nr.html title Boeing and Ball Aerospace Achieve New Milestone for SBSS Program date 2008 04 21 publisher Boeing ref The SBSS development work is being conducted in coordination with the Space Situational Awareness Group in the Space Superiority Systems Wing of the United States Air Force . The commander of the Space Situational Awareness Group believes the SBSS satellite operations center, Will transform Space Situational Awareness by providing a gateway to a responsive, taskable sensor. This capability is key to enabling the event driven operations concept of the future. ref cite web url http www.boeing.com defense space ic sis news 2008 q2 080424c nr.html title Boeing Completes Hardware Installation for SBSS Satellite Operations Center date 2008 04 28 publisher Boeing ref Pathfinder satellite File SBSS1 satellite 01092011.jpg thumb right SBSS 1 2010 048A , the first of the SBSS satellites, passing through Cygnus on Sept 1, 2011 The first pathfinder satellite of the SBSS system was successfully placed into orbit onboard a Minotaur IV rocket on September 25, 2010 ref cite web url http ... launches Minotaur IV publisher 30th Space Wing Public Affairs year 2010 ref . Originally, the launch ... To Launch Site ref was also postponed. ref http news.yahoo.com s ap 20100706 ap on hi te us space ... file media SBSS 2006 10.pdf ref References reflist See also Air Force Space Surveillance System Category Reconnaissance satellites US mil stub de Space Based Space Surveillance hu Space Based Space ... more details
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