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Space

Specialty Definition: Space

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Specialty Definition: Frequency-shift keying

(From Wikipedia, the free Encyclopedia)

Frequency-shift keying (FSK) is frequency modulation in which the modulating signal shifts the output frequency between predetermined values.

Note 1: Usually, the instantaneous frequency is shifted between two discrete values termed the "mark frequency" and "space frequency". This is a noncoherent form of FSK.

Note 2: Coherent forms of FSK exist in which there is no phase discontinuity in the output signal. Synonyms frequency-shift modulation, frequency-shift signaling.

Source: from Federal Standard 1037C and from MIL-STD-188

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Frequency-shift keying."

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Geography

(From Wikipedia, the free Encyclopedia)

Geography is the study of the locational and spatial variation in both physical and human phenomena on Earth. The word derives from the Greek words hê gê ("the Earth") and graphein ("to write", as in "to describe").

Geography is also the title of various historical books on this subject, notably the Geographia by Klaudios Ptolemaios (2nd century).

Geography is much more than cartography, the study of maps. It not only investigates what is where on the Earth, but also why it's there and not somewhere else, sometimes referred to as "location in space". It studies this whether the cause is natural or human. It also studies the consequences of those differences.

History of Geography

The Greekss are the first known culture to actively explore geography as a science and philosophy, with major contributors including Thales of Miletus, Herodotus, Eratosthenes, Hipparchus, Aristotle, Dicaearchus of Messana, Strabo, and Ptolemy. Mapping by the Romanss as they explored new lands added new techniques.

During the Middle Ages, Arabs such as Idrisi, Ibn Battuta, and Ibn Khaldun built on and maintained the Greek and Roman learnings. Following the journeys of Marco Polo, interest in geography spread throughout Europe. During the Renaissance and into the 16th and 17th centuries the great voyages of exploration revived a desire for solid theoretical foundations and accurate detail. The Geographia Generalis by Bernhardus Varenius and Gerardus Mercator's world map are prime examples.

By the 18th century, geography had become recognized as a discrete discipline and became part of a typical university curriculum. Over the past two centuries the quantity of knowledge and the number of tools has exploded. There are strong links between geography and the sciences of geology and botany.

Methods

Spatial interrelationships are key to this synoptic science, and it uses maps as a key tool. Classical cartography has been joined by the more modern approach to geographical analysis, computer-based geographic information systems (GIS).

Geographers use four interrelated approaches:

• Systematic - Groups geographical knowledge into categories that can be explored globally
• Regional - Examines systematic relationships between categories for a specific region or location on the planet.

• Descriptive - Simply specifies the locations of features and populations.
• Analytical - Asks why we find features and populations in a specific geographic area.

Branches

Physical geography

This branch focuses on Geography as an Earth science, making use of biology to understand global flora and fauna patterns, and mathematics and physics to understand the motion of the earth and relationship with other bodies in the solar system. It also includes landscape ecology and environmental geography.

atmosphere -- archipelago -- continent -- desert -- island -- landform -- ocean -- sea -- river -- ecology -- climate -- soil -- geomorphology -- biogeography - Timeline of geography, paleontology

Human geography

The human, or political/cultural, branch of geography - also called anthropogeography focuses on the social science, non-physical aspects of the way the world is arranged. It examines how humans adapt themselves to the land and to other people, and in macroscopic transformations they enact on the world. It can be divided into the following broad categories: economic geography, political geography (including geopolitics), social geography (including urban geography), feminist geography , environmentalism, and military geography.

Countries of the world -- country -- nation -- state -- union -- province -- county -- city -- municipality

Historical geography

This branch seeks to determine how cultural features of the multifarious societies across the planet evolved and came into being. Study of the landscape is one of many key foci in this field - much can be deduced about earlier societies from their impact on their local environment and surroundings.

What's in a name? Historical Geography and the Berkeley School

"Historical Geography" can indeed refer to the recipriocal effects of geography and history on each other. But in the United States, it has a more specialized meaning: This is the name given by Carl Ortwin Sauer of The University of California, Berkeley to his program of reorganizing cultural geography (some say all geography) along regional lines, beginning in the first decades of the 20th Century.

To Sauer, a landscape and the cultures in it could only be understood if all of its influences through history were taken into account: Physical, cultural, economic, political, environmental. Sauer stressed regional specialization as the only means of gaining expertise on regions of the world.

Sauer's philosophy was the principal shaper of American geographic thought in the mid-20th century. Regional specialists remain in academic geography departments to this day. But many geographers feel that it harmed the discipline in the long run: Too much effort was spent on data collection and classification, and too little on analysis and explanation. Studies became more and more area specific as later geographers struggled to find places to make names for themselves. This probably led in turn to the 1950's crisis in Geography which nearly destroyed it as an academic discpline.

Geographic Techniques

• Cartography studies the representation of the Earth's surface with abstract symbols. It can be said, without much controversy, that cartography is the seed from which the larger field of Geography grew. Most geographers will cite a childhood fascination with maps as an early sign they would end up in the field. Although other subdisciplines of geography rely on maps for presenting their analyses, the actual making of maps is abstract enough to be regarded separately.
  Cartography has grown from a collection of drafting techniques into an actual science. Cartographers must learn cognitive psychology and ergonomics] to understand which symbols convey information about the Earth most effectively, and [behavioral psychology] to induce the readers of their maps to act on the information. They must learn geodesy and fairly advanced mathematics to understand how the shape of the Earth affects the distortion of map symbols projected onto a flat surface for viewing.

• Geographic Information Systems deals with the storage of information about the Earth for automatic retrieval by a computer, in an accurate manner appropriate to the information's purpose. In additon to all of the other subdisciplines of geography, GIS specialists must understand computer science and database systems. GIS has so revolutionized the field of cartography that nearly all mapmaking is now done with the assistance of some form of GIS software.

• Geographic quantitative methods deal with numerical methods peculiar to (or at least most commonly found in) geography. In addition to spatial analyses, you are likely to find things like cluster analysis, discriminant analysis, and non-parametric statistical tests in geographic studies.

Related Fields

Urban and Regional Planning

Urban planning and regional planning use the science of geography to assist in determining how to develop (or not develop) the land to meet particular criteria, such as safety, beauty, economic opportunities, the preservation of the built or natural heritage, etcetera. The planning of towns, cities and rural areas may be seen as applied geography although it also draws heavily upon the arts, the sciences and lessons of history. Some of the issues facing planning are considered briefly under the headings of rural exodus, urban exodus and Smart Growth.

Regional Science

In the 1950s the regional science movement arose, led by Walter Isard to provide a more quantitative and analytical base to geographical questions, in contrast to the more qualitative tendencies of traditional geography programs. Regional Science comprises the body of knowledge in which the spatial dimension plays a fundamental role, such as regional economics, resource management, location theory, urban and regional planning, transportation and communication, human geography, population distribution, landscape ecology and environmental quality.

See also

List of countries simple:Geography

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Geography."

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Space

(From Wikipedia, the free Encyclopedia)

The definition of space in physics is contentious. Various concepts used to try to define space have included:

• the structure defined by the set of "spatial relationships" between objects
• a manifold defined by a coordinate system where an object can be located.
• the entity that stops all objects in the universe from touching one another

In classical physics, space is a three-dimensional Euclidean space where any position can be described using three coordinatess. Relativistic physics examines spacetime rather than space; spacetime is modeled as a four-dimensional manifold.

Philosophical questions concerning space include: Is space absolute or purely relational? Does space have one correct geometry, or is the geometry of space just a convention? Historical Eminences who have taken sides in these debates include Isaac Newton (space is absolute), Gottfried Leibniz (space is relational), and Henri Poincaré (spatial geometry is a convention).

Two important thought-experiments connected with these questions are: Newton's bucket argument and Poincaré's disc-world.

See also: Spherical coordinates, Cartesian coordinates, Philosophy of physics

Space is the relatively empty parts of the Universe, outside the atmospheress of planets. It is sometimes called "outer space" to distinguish it from airspace and terrestrial locations.

As Earth's atmosphere has no abrupt cut-off, but rather thins gradually with increasing altitude, there is no definite boundary between the atmosphere and space. In the United States, persons who travel above an altitude of 50 miles (80 kilometers) are designated as astronauts. 400,000 feet (75 miles or 120 kilometers) marks the boundary where atmospheric effects become noticeable during re-entry. The altitude of 100 kilometers or 62 miles is also frequently used as the boundary between atmosphere and space.

See also: Astronomy and Astrophysics; space science; space colonization

The term "inner space" has sometimes been used to describe the contents of the human mind.

See also: psychology

In mathematics, a space is a set, usually with some additional structure.

For examples, see Euclidean space, vector space, normed vector space, Banach space, inner product space, Hilbert space, topological space, uniform space, and metric space.

In some orthographies, a space is a blank area that serves as punctuation to provide interword separation.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Space."

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Space (punctuation)

(From Wikipedia, the free Encyclopedia)

A space is a punctuation convention for providing interword separation in some scripts, including the Latin, Cyrillic, and Arabic.

Not all languages use spaces between words; the ancient Latin and Greek did not. Spaces were not used to separate words until roughly 600-800 AD. (See interword separation for more on the history.) Traditionally, all CJK languages have no space: modern Chinese and Japanese still don't, but modern Korea uses space.

In computer programming, the space corresponds to Unicode and ASCII character 32, or 0x0020.

In programming language syntax, spaces are frequently used to explicitly separate tokens. Aside from this use, spaces and other whitespace are usually ignored by most modern programming languages; Python is one exception.

In word processors and text editorss, if a line on a screen is shorter than the width of the screen or window, then the empty space to the right usually does not correspond with space characters in the file: there is simply a code indicating that the next text should be put on a new line. Thus, the size of the file is not made unnecessarily larger. If there are space characters, one usually does not see the difference; text editors and word processors often have an option to make them visible. Also, if there is a space character, the cursor can move there, otherwise usually not.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Space (punctuation)."

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Space colonization

(From Wikipedia, the free Encyclopedia)

Space colonization, also called space settlement, is the hypothetical permanent autonomous (self sufficient) human habitation of locations outside Earth. It is a major theme in science fiction.

Method

For humans to live permanently outside Earth, the habitat must maintain variables within an appropriate range, ie. homeostasis. The habitat must contain non-human species--for example, microorganisms and crop plants.

The relationship between organisms, their habitat and the non-Earth environment can be:

• Organisms and their habitat fully isolated from the environment
• Changing the environment to become a life-friendly habitat (a process called terraforming)
• Changing organisms to become more compatible with the environment, ie. integrating the habitat into organisms (See also: genetic engineering, transhumanism, cyborg)

A combination of the above is also possible.

Location

The location of colonization can be:

• On a planet, natural satellite or asteroid
• In space, on a stationary space habitat or a mobile spaceship

Location is a frequent point of contention between space colonization advocates.

Planet, natural satellite or asteroid

Mars

Mars is a frequent topic of discussion. Its size and mass is similar to Earth, has large water reserves, and has carbon (locked as carbon dioxide in the atmosphere). It may have gone though similar geogloical and hydrolocial processes as Earth and contian vabluable mineral ores, but this is debated. Equipment is available to extract in-situ resources (water, air, etc.) from the Martian ground and atmosphere.

However, its atmosphere is very thin (7 milibars) and the climate is colder. There is also the problem of native bacteria, which may live on Mars.

The Moon

Due to its proximity and relative familiarity, Earth's Moon is also frequently discussed as a target for colonization. It has the benefits of close proximity to Earth and lower gravity, allowing for easier exchange of goods and services. A major drawback of the Moon is its low abundance of volatiles necessary for life such as hydrogen and carbon. Water ice deposits thought to exist in some polar craters could serve as significant sources for these elements.

Europa

The Artemis Project designed a plan to colonize Europa, one of Jupiter's moons. It would use igloos made of ice refrozen melted by the mircowaves on the surface. For submairne/drill would be use for drilling into the Europan ice crust, as well as any sub-surface ocean. It also discuss use of "air pockets" for human inhabitation.

Space

Space habitat

A space habitat is a space station which is intended as a permanent settlement rather than as a simple waystation or other specialized facility. They would be literal "cities" in space, where people would live and work and raise families. No space habitats have yet been constructed, but many design proposals have been made with varying degrees of realism by both science fiction authors and engineers.

Most of the real work on space habitats was carried out in the 1970s by workshops led by Gerard K. O'Neill in the post-Apollo highs at NASA. Several designs were studied, some in depth, with sizes ranging from 1,000 to 10,000 people. Attempts were made to make the habitats as self-supporting as possible, but all of the designs relied on regular shipments from Earth or the Moon, notably for volatiles.

One problem with the design that was not considered in any real depth is why any of them would be needed. The stated problem was to house workers needed for the construction of solar power satellites, which they predicted would require a peak of about 25,000 workers. However if this was the purpose, the habitat designs were certainly not utilitarian; they all contained housing for complete families, huge open spaces, and considerable parkland. An oil platform would appear to be a better model for such purposes. The workshops appeared to work in reverse, inventing the "solution", and then casting about for a need.

Designs proposed include:

• Bernal sphere - "Island One"
• Stanford torus - "Island Two"
• O'Neill cylinder - "Island Three"

Spaceship

A colony ship would be similar to a space habitat, except with major propulsion capabilities and independent power generation.

Concepts proposed in hard science fiction include:

• Generation ship, hypothetical starship that would travel much slower than light between stars, with the crew going through multiple generations before the journey is complete
• Sleeper ship, hypothetical spaceship in which most or all of the crew spend the journey in some form of hibernation or suspended animation

Justification

In 2001, the space news website SPACE.com asked Freeman Dyson, J. Richard Gott and Sid Goldstein for reasons why some humans should live in space. Their respective answers [1] were:

• "To Spread Life and Beautify the Universe"
• "To Ensure the Survival of Our Species"
• "To Make Money and Save the Environment"

Advocacy

There are several space advocacy organizations. The major ones are listed below:

• The National Space Society is an organization with the vision of "people living and working in thriving communities beyond the Earth." It was founded in 1960. [1]
• The Mars Society promotes Robert Zubrin's Mars Direct plan and the settlement of Mars. [1]
• The Space Frontier Foundation promotes strong free market, capitalist views about space development. [1]
• The Living Universe Foundation has a detailed plan in which the entire galaxy is colonized. [1]
• The Artemis Project is planning to set up a private lunar surface station. [1]
• The Space Studies Institute was founded by Gerard K. O'Neill to fund the study of space habitats. [1]
• The Planetary Society is the largest space interest group, but has an emphasis on robotic exploration and the search for extraterrestrial life. [1]

Fictional depictions

These are all films and books that display famous depictions of space colonies of Earth.

• 2001: A Space Odyssey (in space)
• Alien, Aliens, and Alien 3 (on a few different planets)
• Cowboy Bebop (throughout the Solar System)
• Martian Chronicles, The (on Mars)
• Solaris (around the star Solaris)
• Total Recall (on Mars)

Related articles

• Ocean colonization
• Underground city
• Underground colonization

External links

• HobbySpace: Life in Space: Section C: Colonies, Habitats, Space Industry, etc Extensive collection of links
• Orbital Space Settlements Al Globus (NASA) advocates space habitats, not planetary habitats.
• Space Development: The Case Against Mars by K. Eric Drexler. A 1985 article with arguments against Matian colonization.
• Space Settlements: A Design Study Authored by the participants of "The 1975 Summer Faculty Fellowship Program in Engineering Systems Design" under the sponsorship of NASA and American Society for Engineering Education. Proposal for a space habitat with 10,000 people.
• Warm-Blooded Plants and Freeze-Dried Fish Freeman Dyson predicts that space colonization will only be affordable after a hundred years; and that biotechnology, not propulsion, will be the enabling factor.
• Space and Human Survival: My Views on the Importance of Colonizing Space Sylvia Engdahl discusses the "critical stage" where a level of technology allows both space colonization and human extinction.
• The Political Economy of Very Large Space Projects John Hickman argues that only government can afford the high initial investment for very large space development projects.
• PERMANENT (Projects to Employ Resources of the Moon and Asteroids Near Earth in the Near Term) is guide to websites about astroid settlement.

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Space colonization."

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Space exploration

(From Wikipedia, the free Encyclopedia)

Space exploration is the physical exploration of outer-earth objects and generally anything that involves the technologies, science, and politics regarding space endeavors.

The idea of sending an object to space was conceived in the minds of many science fiction authors hundreds of years before it was actually feasible. Some of these works even included various descriptions of exactly how that would be done. During the 20th century, with the development of adequate propulsion technologies, stronger and lighter materials and other technological and scientific breakthroughs, the idea of outer-earth missions was no longer a dream, but a viable practice.

The first successful space launch was of the Russian unmanned Sputnik I mission on October 4, 1957. Manned space exploration began with the orbital flight of Yuri Gagarin in April, 1961.

See also:

• Space Race
• Manned space missions
  • Vostok program
  • Mercury program
  • Soyuz program
  • Gemini program
  • Apollo program and others
  • Skylab
  • Space Shuttle program
  • Shenzhou spacecraft
• Unmanned space missions
  • Mariner program and others
  • Surveyor program
• Astronauts
• Space stations
• Space tourism
• Space colonization

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Space exploration."

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Space science

(From Wikipedia, the free Encyclopedia)

Space is the relatively empty parts of the universe, outside the atmospheress of planets. It is sometimes called "outer space" to distinguish it from airspace and terrestrial locations. It may be considered to start at a height of ca. 100 km above the Earth's surface. For the term in general see Space.

Space science, or the space sciences, includes several major fields, such as:

• Astronomy and Astrophysics
• Exobiology
• Microgravity environment
• Plasma physics

• Space transport
  • Rocket propulsion
  • Rocket launch technology
  • Interplanetary travel
  • Interstellar travel
  • Spacecraft propulsion

• Space exploration
  • Unmanned space missions
  • Manned space missions

• Space colonization

In addition, space sciences impact or are related to many other fields, from the biology of organisms in space environments to the geology of other bodies and planets, as well as nuclear physics in interstellar space and inside stars.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Space science."

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Spacetime

(From Wikipedia, the free Encyclopedia)

In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional manifold called spacetime (alternatively, space-time; see below). A point in spacetime may be referred to as an event. Each event has four coordinates (t, x, y, z).

Reference frame

Just as the x, y, z coordinates of a point depend on the axes one is using, so distances and time intervals, invariant in Newtonian physics, may depend on the reference frame of an observer, in relativistic physics. See length contraction and time dilation. This is the central lesson of special relativity.

The central lesson of general relativity is that spacetime cannot be a fixed background, but is rather a network of evolving relationships.

A spacetime interval between two events is the frame-invariant quantity analogous to distance in Euclidean space. The spacetime interval s along a curve is defined by

where c is the speed of light (some people flip the signs of the equation). A basic assumption of relativity is that coordinate transformations have to leave intervals invariant. Intervals are invariant under Lorentz transformations.

The spacetime intervals on a manifold define a pseudo-metric called the Lorentz metric. This metric is very similar to distance in Euclidean space. However, note that whereas distances are always positive, intervals may be positive, zero, or negative. Events with a spacetime interval of zero are separated by the propagation of a light signal. Events with a positive spacetime interval are in each other's future or past, and the value of the interval defines the proper time measured by an observer travelling between them. Spacetime together with this pseudo-metric makes up a pseudo-Riemannian manifold.

One of the simplest interesting examples of a spacetime is R⁴ with the spacetime interval defined above. This is known as Minkowski space, and is the usual geometric setting for Special Relativity. In contrast, General Relativity says that the underlying manifold will not be flat, if gravity is present, and thus it calls for the use of spacetime rather than Minkowski space.

Strictly speaking one can also consider events in Newtonian physics as a single spacetime. This is Galilean-Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a spacetime can be decomposed unarbitrarily, which is not possible in the general case.

Some general facts about spacetimes

A compact manifold can be turned into a spacetime if and only if its Euler characteristic is 0.

Any non-compact 4-manifold can be turned into a spacetime.

Many spacetimes have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compact spacetime has closed timelike curves, which violate our usual ideas of causality. For this reason, mathematical physicists usually consider only restricted subsets of all the possible spacetimes. One way to do this is to study "realistic" solutions of the equations of General Relativity. Another way is add some additional "physically reasonable" but still fairly general geometric restrictions, and try to prove interesting things about the resulting spacetimes. The latter approach has lead to some important results, most notably the Penrose-Hawking singularity theorems.

In mathematical physics it is also usual to restrict the manifold to be connected and Hausdorff. A Hausdorff spacetime is always paracompact.

Is Spacetime Quantized?

Current research is focused on the nature of spacetime at the Planck scale. Loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized spacetime with agreement on the order of magnitude. Loop quantum gravity even makes precise predictions about the geometry of spacetime at the Planck scale.

Space-time vs. Spacetime

Examples of use of spacetime:

• Weisstein's encyclopedia
• D. J. Griffiths' Introduction to Electrodynamics (Upper Saddle River, N. J.: Prentice-Hall, 1989)
• numerous books with spacetime in title
  • E. F. Taylor and J. A. Wheeler, Spacetime Physics (San Francisco: W. H. Freeman, 1966)

• Caltech class: "Spacetime 101"
• .edu matches online are almost exclusively for spacetime

Examples of use of space-time:

• Brehm & Mullin, Introduction to the Structure of Matter (ISBN 047160531X)
• Hawking & Ellis, The large-scale structure of space-time (ISBN 0521099064)
• Merriam-Webster
• space-time about four times as many hits as spacetime on AltaVista

Related concepts

Galilean transformation | Lorentz transformation | Minkowski space | Lorentz invariance | Manifold | Metric space | Gravity | Four-vector

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Spacetime."

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Topological space

(From Wikipedia, the free Encyclopedia)

Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in all branches of modern mathematics and can be seen as a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology.

History

See topology.

Formal definition

Formally, a topological space is a set X together with a collection T of subsets of X (i.e., T is a subset of the power set of X) satisfying the following axioms:

1. The empty set and X are in T.
2. The union of any collection of sets in T is also in T.
3. The intersection of any pair of sets in T is also in T.

The set T is called a topology on X. The sets in T are referred to as open sets, and their complements in X are called closed sets. The elements of X are often called points. Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points.

Continuous functions

A function between topological spaces is said to be continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "breaks" or "separations" in the function. A homeomorphism is a bijective mapping that is continuous and whose inverse is also continuous. Two spaces are said to be homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.

The category of all topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in all mathematics. The attempt to classify the objects of this category by invariants has motivated and generated entire areas of research, such as homotopy theory, homology theory, and K-theory, to name just a few.

Alternative definitions

There are many other equivalent ways to define a topological space. (In other words, each of the following defines a category equivalent to the category of topological spaces above.)

• Using de Morgan's laws, the axioms defining open sets become axioms defining closed sets:

The empty set and X are closed.

The intersection of any collection of closed sets is also closed.

The union of any pair of closed sets is also closed.

• The Kuratowski closure axioms determine the closed sets as the fixed points of an operator on the power set of X.

• A neighbourhood of a point x is any set that contains an open set containing x. The neighbourhood system at x consists of all neighbourhoods of x. A topology can be determined by a set of axioms concerning all neighbourhood systems. Equivalently, a topology can be determined by a nearness relation between sets and points.

• A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X the set of its accumulation points is specified.

Examples of topological spaces

• The set of real numbers R is a topological space: the open sets are generated by the basis of open intervals. This means a set is open if it is the union of (possibly infinitely many) open intervalss. This is in many ways the most basic topological space and the one that guides most of our human intuition. However, relying on the real line as an intuitive guide for the general concept of topological space can often be dangerous.
• More generally, the Euclidean spaces Rⁿ are topological spaces, and the open sets are generated by open balls.
• Any metric space turns into a topological space if define the open sets to be generated by the set of all open balls. This includes useful infinite-dimensional spaces like Banach spaces and Hilbert spaces studied in functional analysis.
• The reals can also be given the upper-limit topology. Here, the open sets consist of the empty set, the whole real line, and all sets generated by half-open intervals of the form (a,b]. This topology on R is strictly larger than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from below in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.
• Every manifold is a topological space.
• Every simplex is a topological space. Simplexes are convex objects that are very useful in computational geometry. In 0, 1, 2 and 3 dimensional space the simplexes are the point, line segment, triangle and tetrahedron, respectively.
• Every simplicial complex is a topological space. A simplicial complex is made up of many simplices. Many geometric objects can be modeled by simplicial complexes -- see also Polytope.
• The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety. On Rⁿ or Cⁿ the closed sets of the Zariski topology are the solution sets of systems of polynomial equations.
• A linear graph is a topological space that generalises many of the geometric aspects of graphss with vertices and edges.
• Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
• Any set can be given the discrete topology in which every set is open. The only convergent sequences or nets in this topology are those that are eventually constant.
• Any set can be given the trivial topology in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique.
• Any infinite set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T₁ topology on the set.
• If Γ is an ordinal number, then the set [0, Γ] is a topological space, generated by the intervals (a,b], where a and b are elements of Γ.

Constructing new topological spaces from given ones

• Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset.
• For any nonempty collection of topological spaces, the product can be given the product topology. For finite products, the open sets are generated by the products of open sets.
• A quotient space is defined as follows. If f: X → Y is a function and X is a topological space, then Y gets a topology where a set is open if and only if its inverse image is open. A common example comes from an equivalence relation defined on the topological space X: the map f is then the natural projection on the set of equivalence classes.
• The Vietoris topology on the set of all non-empty subsets of a topological space X is generated by the following basis: for every n-tuple U₁,....,U_n of open sets in X we construct a basis set consisting of all subsets of the union of the U_i which have non-empty intersection with each U_i.

Classification of topological spaces

Topological spaces can be broadly classified according to their degree of connectedness, their size, their degree of compactness and the degree of separation of their points and subsets. A great many terms are used in topology to achieve these distinctions. These terms and definitions are collected together in the Topology Glossary.

Topological spaces with algebraic structure

It is almost universally true that all "large" algebraic objects carry a natural topology which is compatible with the algebraic operations. In order to study these objects, one typically has to take the topology into account. This leads to concepts such as topological groups, topological vector spaces and topological rings.

See also Algebraic topology.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Topological space."

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Abbreviations & Acronyms: Space

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Synonyms within Context: Space

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Modern Usage: Space

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Commercial Usage: Space

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Image Slideshow: Space

Photos: Space More pictures...

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Photo Album: Space

Thumbnail	Description & Credit	Thumbnail	Description & Credit
	Shown is the Boy Scout Building located at Cedar Lane and Wisconsin Avenue in which the NIH leases office space. Credit: Unknown photographer/artist.		The photo shows a model of the Children's Inn in an overhead shot from the front. The model includes the Inn itself, a parking lot and surrounding space and trees. The Children's Inn houses families of children undergoing treatment at the National Institutes of Health (NIH). Credit: Unknown photographer/artist.
	Histopathology of lung shows numerous extracellular yeasts of Cryptococcus neoformans within an alveolar space. Yeasts show narrow-base budding and characteristic variation in size. Credit: CDC.		Seattle at nightfall, from top of Space Needle. Credit: CDC.
	Space Shuttle Atlantis EVA. Credit: NASA.		MK-III Space Suit. Credit: NASA.
	Fall Foliage from Space. Credit: NASA.		Australia from Space in All Her Olympic Glory. Credit: NASA.
	Endeavour is Delivered to the Kennedy Space Center. Credit: NASA.		Space Shuttle Main Engine (SSME) Test Firing. Credit: NASA.
Source: pictures compiled by the editor from various references; see picture credits.

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Digital Photo Gallery: Space
 

"Space Needle at Night" by Heather Laidlaw Commentary: "Some neon by the Space Needle in Seattle."	"Space Needle" by Sarah Benton Commentary: "Space Needle from a different view."
Source: photographs selected by the editor, with permission from the photographers.

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Sounds Captioned with "Space".

Play	Caption	Play	Caption
	Space mystery style television show excerpt.		Vapid space sound.
	Space bounce.
Source: compiled by the editor from various references; see credits.

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Familiar Quotations: Space

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Historic Usage: Space

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Use in Literature: Space

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Non-Fiction Usage: Space