Copyright © Philip M. Parker, INSEAD. Terms of Use.

Open

Specialty Definition: Open

Top     

Specialty Definition: Continuous revelation

(From Wikipedia, the free Encyclopedia)

In The Church of Jesus Christ of Latter-day Saints, continuous revelation is the principle that God or his divine agents still continue to communicate to mankind. This communication can be manifest in innumerable ways: the innumerable influences of the Holy Ghost; vision; visitation of divine beings; and others. By such means God guides his followers to salvation and without such His followers will eventually form their beliefs or practices after a god of their own making. The founder of the Church, Joseph Smith explained the importance and necessity of continuous revelation:

God said, "Thou shalt not kill;" at another time He said, "Thou shalt utterly destroy." This is the principle on which the government of heaven is conducted-by revelation adapted to the circumstances in which the children of the kingdom are placed. Whatever God commands is right, no matter what it is, although we may not see the reason thereof till long after the events transpire . . . As God has designed our happiness-and the happiness of all His creatures, he never has-He never will institute an ordinance or give a commandment to His people that is not calculated in its nature to promote that happiness which He has designed, and which will not end in the greatest amount of good and glory to those who become the recipients of his law and ordinances. (Teachings of the Prophet Joseph Smith, pp.256-7.)

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

Top     

Open (poker)

(From Wikipedia, the free Encyclopedia)

In poker, the act of making the first non-zero bet in a betting round is called opening the round. On the first betting round, it is also called opening the pot. Some games may have special rules about opening a round that may not apply to other bets (for example, they may have a betting structure that specifies different allowable amounts for opening than for other bets, or they may require a player to hold certain cards to open).

The term is also used to describe a category of poker game in which some cards held by individual players are visible to all other players, either by being dealt face up or by being exposed during play (but before showdown). Most forms of stud poker are open poker (blind stud games are an exception). Most forms of draw poker, in contrast, are closed games because no player's cards are seen until showdown (draw games with a rollout are an exception). Most community card games like Texas hold'em are considered closed as well, because the only cards exposed before showdown belong to everyone; the individual players' cards are never seen until showdown.

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

Top     

Open circuit

(From Wikipedia, the free Encyclopedia)

An open circuit is when there is nothing connected to a load and no current can flow. This can be represented by a resistance/impedance that is equal to infinity.

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

Top     

Open class word

(From Wikipedia, the free Encyclopedia)

Open class words belong to a list of morphemes which can be added to by absorbing new words, such as technical terms, slang, as well as adoptions and adaptations of foreign words.

The list of groups of words that belong to the open class type include:

• noun
• main verb, (however not auxiliary verbs)
• adjective
• adverb
• interjection

Interjections particularly, are formed as new words standing in for sounds, and are added not only from technical backgrounds, but also from sources such as comicss and subtitling. It is in these that one will encounter the noises of motor revving, sirens, mechanical sounds and violence, continuously being updated. Examples here are: vroom!, va-va-voom!, zonk!, grrh!, and so on.

Closed-class words such as prepositions, conjunctions, pronouns, etc., are words belonging to a class to which no new words can be added readily.

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

Top     

Open cluster

(From Wikipedia, the free Encyclopedia)

An open cluster is a group of stars that were born at the same time from a molecular cloud, and are still near to each other. They are also called galactic clusters since they exist within the galaxy's disk.

Open clusters are usually young (in astronomical terms), and so contains many hot and luminous stars. This makes open clusters visibile from large distances and one of the preferred objects by amateur astronomers. The "parent" molecular cloud is sometimes still associated with the cluster, which illuminates parts of the cloud that are then visibile as one or more nebulae. .

All the stars in an open cluster have more or less the same age and the same chemical composition, so any difference between them is solely due to their mass. Most open clusters are dominated by their O-type and B-type giant blue stars, which are very luminous but short-lived. Analysing the light from an open cluster one can estimate its age, looking at ratio between blue, yellow and red stars. The more blue stars there are, the younger the cluster is. The uniformity between the cluster's stars makes them a perfect test for stellar evolution models, because when comparing one star to the other, most of the variable parameters are now fixed. Testing the model is therefore easier.

The closest open cluster is in Ursa Major, or to be more correct, it is Ursa Major. Most of the stars in the famous asterism are members of an old and mostly dispersed open cluster. Sirius is a former member of this cluster and our sun is in the outskirts of what is called The Ursa Major Stream, a group of stars that are all ex-members of the Ursa Major Cluster spanning over a thousand light years in space. Our Sun is not a member, however, just passing through. Considering its location in the galaxy our Sun has a very strange velocity, we probably had a close encounter with another star a few billion years ago which gravitationally accelerated the Sun and the solar system.

Stars inside an open cluster are at first tighly packed, moving at the same speed around the center of the Galaxy. After half a billion years or so, a classic open cluster such as the Pleiades or the Hyades (both in Taurus) tends to be disturbed by external factors (such as molecular clouds passing by), setting its stars moving at slightly different speeds and so causing them to drift apart exactly like the one in Ursa Major has done. When this happens, the cluster becomes a stream of stars, not close enough to be a cluster but all related and moving in very similar directions at similar speeds.

After a billion or so years, the cluster is totally lost. Some stars will be on the far side of the galaxy, some on the near. The sun's original cluster is like this, there is no way to tell which are former members and which just happened to have formed at the same time but somewhere else.

The exact timeline of this evolution may vary according to the cluster start density: more tighly packed clusters will survive for longer times, but no known cluster has survived more than a few billion years.

External Link

• Open Star Clusters, SEDS Messier pages

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

Top     

Open map

(From Wikipedia, the free Encyclopedia)

In topology, an open map is a function between two topological spaces which maps open sets to open sets.

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

Top     

Open set

(From Wikipedia, the free Encyclopedia)

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, x can't be on the edge of U.

As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and wiggle a tiny bit in the positive direction, you will be outside of (0,1].

Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers. Note also that "open" is not the opposite of "closed". First, there are sets which are both open and closed (called clopen sets); in R and other connected spaces, only the empty set and the whole space are clopen, while the set of all rational numbers smaller than √2 is clopen in the rationals. Also, there are sets which are neither open nor closed, such as (0,1] in R.

Definitions

The concept of open sets can be formalized in various degrees of generality.

Euclidean space

A subset U of Euclidean n-space Rⁿ is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rⁿ whose Euclidean distance from x is smaller than ε, y also belongs to U.

Intuitively, ε measures the size of the allowed "wiggles".

Metric spaces

A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U.

This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological spaces

In topological spaces, the concept of openness is taken to be fundamental. One starts with an arbitrary set X and a family of subsets of X satisfying certain properties that every "reasonable" notion of openness is supposed to have. (Specifically: the union of open sets is open, the finite intersection of open sets is open, and in particular the empty set and X itself are open.) Such a family T of subsets is called a topology on X, and the members of the family are called the open sets of the topological space (X,T).

This generalises the metric space definition: If you start with a metric space and define open sets as before, then the family of all open sets will form a topology on the metric space. Every metric space is hence in a natural way a topological space. (There are however topological spaces which are not metric spaces.)

Uses

Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A.

Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The map f is called open if the image of every open set in X is open in Y.

Manifolds

A manifold is called open if it is a manifold without boundary and if it is not compact. This notion differs somewhat from the openness discussed above.

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

Top     

Open source

(From Wikipedia, the free Encyclopedia)

Open-source computer software is software whose source code is either in the public domain or, more commonly, is copyrighted by one or more persons/entities but licensed to all according to an open-source license. Such a license grants permission to use and redistribute the software, as well as to modify its source code and distribute modified versions, with at most minor restrictions (such as a requirement to preserve the authors' names and copyright statement in the code).

These are rights for users of the software. An open-source license itself does not necessarily require that the software, or its source, initially has to be freely (in both senses of the word) available on the Internet. Most popular open-source software is, however.

The term open source in common usage may also refer to any software with publicly available source code, regardless of its license, but this usage provokes strong disapproval from the open source community. Examples of such "disclosed source" software include some versions of Solaris and PGP.

"Open source" and "Free software"

In the strict definition, the term "open source" is distinct from "free software," and it should only be applied to software that meets the terms of the Open Source Definition (see also the Free Software Foundation's (FSF) Free software definition). The decision to adopt the term "open source", suggested by Christine Peterson of the Foresight Institute, was based partly on the confusion caused by the dual meaning of the word "free"; the FSF intended the word to mean "free speech, not free beer," but nevertheless, free software came to be associated with zero cost, a problem which was exacerbated by the fact that a great deal of it is, in fact, free of charge. It was hoped that the usage of the newer term "open source" would eliminate such ambiguity, and would also be easier to "market" to business users (who might mistakenly associate "free software" with anti-commercialism). Since its introduction, however, the "open source" label has been criticized for fostering an ambiguity of a different kind: that of confusing it for mere availability of the source, rather than the freedom to use, modify, and redistribute it.

The Free Software Definition is slightly more restrictive than the Open Source Definition; as a consequence of this, free software is open source, but open source software may or may not be "free." In practice, nearly all open-source licenses also satisfy the FSF's free-software definition, and the difference is more a matter of philosophical emphasis. (One of the few counter-examples was an early version the Apple Public Source License, which was considered open source but not free because it did not allow private modified versions; this restriction was later removed.) For instance, software distributed under both the GPL and BSD licenses are considered both free and open source (the original BSD License had terms legally incompatible with the GPL, but this practical difficulty is a separate issue from its free-ness). Confusion about the distinctions between free and open source software is the source of some misunderstanding, particularly in the mass media where the two terms are often applied interchangeably.

The open source movement

The open-source movement is a large movement of programmers and other computer users to give easy access to computer software. It grew out of the Free software movement, and the line between the two is somewhat blurry. Mostly, the Free software movement is based upon political and philosophical ideals (sometimes referred as hacker culture), while open source proponents tend to focus on rather pragmatic matters. Both groups assert that this more open style of licensing allows for a superior software development process, and therefore that pursuing it is in line with rational self-interest. Free software advocates, however, would argue that "freedom" is a paramount merit that one should prefer (or at least weigh heavily) even in cases where proprietary software has some superior technical features.

Proponents of the open source development methodology claim that it is superior in a number of ways to the closed source method. Stability, reliability, and security are frequently cited as reasons to support open source. One successful application of the open source model is the Linux operating system, which is renowned for its stability and security characteristics. Among the works that explore and justify open source development is a series of works by Eric S. Raymond which includes The Cathedral and the Bazaar and Homesteading the Noosphere.

Open source advocates point out that as of the early 2000s, at least 90 percent of computer programmers are employed not to produce software for direct sale, but rather to design and customize software for other purposes, such as in-house applications. According to advocates, this statistic implies that the value of software lies primarily in its usefulness to the developer or developing organization, rather than in its potential sale value, and that consequently there is no compelling economical reason to keep source code secret from competitors.

Open Source advocates

• Bruce Perens, Eric Raymond, Linus Torvalds, Paul Vixie, Alan Cox, Tim O'Reilly
• Russell Pavlicek, author of the book Embracing Insanity

Projects and Organizations

• Debian, FreeBSD, Mozilla, NetBSD, OpenBSD, OpenOffice.org, Open Source Initiative, OSDN, Slackware

Companies Involved in Open Source Development

• Apple, Hewlett-Packard, IBM, Red Hat, Sun Microsystems

Examples of Open Source Licenses

• BSD license, GNU General Public License, GNU Lesser General Public License, MIT License, Apache Software License, Mozilla public licence

For a more extensive list, see Open source license.

Examples of Open Source Software

• Apache, Linux, BSD, Mozilla, GNU Emacs, TeX, VIM, XFree86, the GIMP, PHP, Zope, KDE, Gnome, OpenOffice.org

For a more extensive list, see List of open-source software packages.

Related topics

• Richard Stallman is a central figure in the rival (but cooperative) free software movement, which has a significantly different philosophical basis; Stallman does not want his name associated with the term open source.
• open content for non-programming open source projects

See also: Halloween documents, Open Cola, SourceForge, GNU Savannah, Open Law project, Gift economy

External links

• EU-funded report on the use of open-source software
• Open Source HealthCare Alliance
• Open Sources: Voices from the Open Source Revolution - an online book containing essays from prominent members of the open source community
• Open Source Initiative OSI - a list of available licenses
• The Institutional Design of Open Source Programming: Implications for Addressing Complex Public Policy and Management Problems
• The Asian Open Source Center - has a lot of information on open source in Asia, much of it stored in a Wiki similar to Wikipedia
• W3C Open Source Software
• A Case Against Open Source, by Mathias Strasser, 2001, from Stanford Technology Law Review

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

Top     

Open standard

(From Wikipedia, the free Encyclopedia)

Open standards are publicly available specifications for enhancing compatibility between various hardware and software components. Open standards allow anybody with the technical know-how and the necessary equipment to implement solutions which work together with those of other vendors.

Examples of open standards

Hardware:

• ISA
• PCI
• AGP

Software:

• HTML
• SQL
• IP
• TCP

In 2002 and 2003 there was some controversy about using Reasonable And Non-Discriminatory (RAND) licencing for the use of patented technology in web standards. Bruce Perens and others have argued that the use of patents restricts who can implement a standard to those able or willing to pay for the use of the patented technology. The requirement to pay some small amount per user, is often an insurmountable problem for free software or open source implementations which can be redistributed by anyone. Royalty free (RF) licensing is preferred by Open Source adepts. The GNU GPL license includes a section that enjoins any one who distributes a program released under the GPL from enforcing patents on subsequent users of the software or derivative works.

Quotes

• EU Commissioner Erkki Liikanen: "Open standards are important to help create interoperable and affordable solutions for everybody. They also promote competition by setting up a technical playing field that is level to all market players. This means lower costs for enterprises and, ultimately, the consumer." (World Standards Day, 14 October, 2003)

External links

• Text of Liikanen's speech

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

Top     

Topology glossary

(From Wikipedia, the free Encyclopedia)

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and definitions that are fundamental to a broad range of areas. See the article on topological spaces for basic definitions and examples, and see the article on topology for a brief history and description of the subject area.

The following articles may also be useful. These either contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The list of general topology topics will also be very helpful.

• Compact space
• Connected space
• Continuity (topology)
• Metric space
• Separated sets
• Separation axiom
• Uniform space

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.

• Accessible. See T₁.

• Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.

• Base. A set of open sets is a base (or basis) for a topology if every open set in the topology is a union of sets in the base. The topology generated by a base is the smallest topology containing the base elements; this topology consists of all unions of elements of the base.

• Basis. See Base.

• Borel algebra. The Borel algebra on a space X is the smallest σ-algebra containing all the open sets.

• Borel set. A Borel set is an element of a Borel algebra.

• Boundary. The boundary of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement.

• Cauchy sequence. A sequence {x_i} in a metric space M with metric d is called a Cauchy sequence (or Cauchy for short) if for every positive real number r, there is an integer N such that for all integers m and n greater than N, the distance d(x_m, x_n) is less than r.

• Clopen. A set is clopen if it is both open and closed.

• Closed set. A set is closed if its complement is a member of the topology.

• Closed function. A function from one space to another is closed if the image of every closed set is closed.

• Closure. The closure of a set is the intersection of all closed sets which contain it. It is the smallest closed set containing the original set.

• Compact. A space is compact if every open cover has a finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.

• Complete. A metric space is complete if every Cauchy sequence converges.

• Completely metrizable/completely metrisable. See Topologically complete.

• Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods.

• Completely normal Hausdorff. A completely normal Hausdorff space (or T₅ space) is a completely normal T₁ space. (A completely normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Completely normal Hausdorff spaces are always normal Hausdorff.

• Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are functionally separated.

• Completely regular Hausdorff. See Tychonoff.

• Completely T₃. See Tychonoff.

• Component. See connected component.

• Connected. A space X is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.

• Connected component. A connected component of a space is a maximal connected subspace. The connected components of a space form a partition of that space.

• Continuous. A function from one space to another is continuous if the preimage of every open set is open.

• Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.

• Countably compact. A space is countably compact if every countable open cover has a finite subcover.

• Cover. A collection {U_i} of sets is a cover (or covering), if their union is the whole space. An open cover is a cover consisting of open sets.

• Covering. See Cover.

• Dense. A dense set is a set that meets every nonempty open set in the space. Equivalently, a set is dense if its closure is the whole space.

• Discrete topology. See Discrete space.

• Discrete space. A space X is discrete if every set is open. We say that X carries the discrete topology.

• Entourage. See Uniform space.

• F_σ set. An F_σ set is a countable union of closed sets.

• First category. See Meagre.

• First-countable. A space is first-countable if every point has a countable local base.

• Functionally separated. Two sets A and B in a space X are functionally separated if there is a continuous function from X into the interval [0,1] with the property that A is mapped to 0 and B is mapped to 1.

• G_δ set. A G_δ set is a countable intersection of open sets.

• Hausdorff. A space is Hausdorff (or T₂) if every two distinct points have disjoint neighbourhoods. Hausdorff spaces are always T₁.

• Hereditary. A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it. For example, second-countability is a hereditary property.

• Homeomorphism. A homeomorphism from a space X to a space Y is a bijective map f : X → Y such that f and f ^-1 are continuous. The spaces X and Y are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.

• Homogeneous. A space X is homogeneous if for every x and y in X there is a homeomorphism f : X -> X such that f(x) = y. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.

• Homotopic maps. Two continuous maps f, g : X -> Y are homotopic if there is a continuous map H: X× [0,1] → Y, such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X. Here, the space X × [0,1] is given the usual product topology. The function H is called a homotopy between f and g.

• Indiscrete space. See Trivial topology.

• Indiscrete topology. See Trivial topology.

• Interior. The interior of a set is the union of all open sets contained in it. It is the largest open set contained in the original set.

• Isolated point. A point x is an isolated point if the singleton {x} is open.

• Kolmogorov. See T₀.

• Kuratowski closure axioms. The Kuratowski closure axioms are a set of axioms satisied by the closure operator:

Isotonicity: Every set is contained in its closure.

Idempotence: The closure of the closure of a set is equal to the closure of that set.

Preservation of binary unions: The closure of the union of two sets is the union of their closures.

Preservation of nullary unions: The closure of the empty set is empty.

• Limit point. A point x in X is a limit point of a subset S if every open set containing x also contains a point of S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself.

• Lindelöf. A space is Lindelöf if every open cover has a countable subcover.

• Local base. A set B of neighbourhoods of a point x of a topological space X is a local base (or local basis, neighbourhood base, neighbourhood basis) at x if every neighbourhood of x contains some member of B.

• Local basis. See Local base.

• Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Locally compact Hausdorff spaces are always Tychonoff.

• Locally connected. A space is locally connected if every point has a local base consisting of connected sets.

• Locally finite. A collection of subsets of a space is locally finite if every point has a neighbourhood which meets only finitely many of the subsets.

• Locally metrizable/Locally metrisable. A space is locally metrizable if every point has a metrizable neighbourhood.

• Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.

• Meagre. If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable union of nowhere dense sets. If A is not meagre in X, A is sometimes said to be of second category in X.

• Metric. See Metric space.

• Metric space. A metric space is a set M equipped with a function d : M × M → R satisfying the following conditions for all x, y, and z in M:

d(x, y) ≥ 0

d(x, x) = 0

if   d(x, y) = 0   then   x = y     (identity of indiscernibles)

d(x, y) = d(y, x)     (symmetry)

d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality)

The function d is called a metric on M.

• Metrizable/Metrisable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable.

• Neighbourhood/Neighborhood. A neighbourhood of a set S is a set containing an open set which in turn contains the set S. (Note that the neighbourhood itself need not be open.) A neighbourhood of a point p is a neighbourhood of the singleton set {p}.

• Neighbourhood base. See Local base.

• Neighbourhood basis. See Local base.

• Net. A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (x_α), where α is in an index variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers with the usual ordering.

• Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.

• Normal Hausdorff. A normal Hausdorff space (or T₄ space) is a normal T₁ space. (A normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Normal Hausdorff spaces are always Tychonoff.

• Nowhere dense. A nowhere dense set is a set whose closure has empty interior.

• Open cover. See Cover.

• Open set. A set is open if it is a member of the topology.

• Open function. A function from one space to another is open if the image of every open set is open.

• Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.

• Partition of unity. A partition of unity of a space X is a set of continuous functions from X to [0,1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.

• Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected.

• Point. This term is often used to refer to elements of the topological space.

• Polish. A space is called Polish if it is metrizable with a separable and complete metric.

• Product topology. If {X_i} is a collection of spaces and X is the (set-theoretic) product of {X_i}, then the product topology on X is the weakest topology for which all the projection maps are continuous.

• Punctured neighbourhood/Punctured neighborhood. A punctured neighbourhood of a point p is a neighbourhood of p, minus {p}. For instance, the interval (-1,1) = {x : -1 < x < 1} is a neighbourhood of 0 in the real line, so the set (-1,0) ∪ (0,1) = (-1,1) - {0} is a punctured neighbourhood of 0.

• Quotient space. If X and Y are spaces and f : X → Y is any function, then the quotient space on Y induced by f is the weakest topology for which f is continuous. The most common example of this is to consider an equivalence relation on X, with Y the set of equivalence classes and f the natural projection map.

• Refinement. A cover K is a refinement of a cover L if every member of K is a subset of some member of L.

• Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.

• Regular Hausdorff. A space is regular Hausdorff (or T₃) if it is a regular T₀ space. (A regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.)

• Residual. If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X.

• Second category. See Meagre.

• Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

• Separable. A space is separable if it has a countable dense subset.

• Separated. Two sets A and B are separated if each is disjoint from the other's closure.

• Sierpinski space. Let S = {0,1}. Then T = is a topology on S, and the resulting space is called Sierpinski space. The Sierpinski space is the simplest example of a space that does not satisfy the T₁ axiom.

• Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S¹ → X is homotopic to a constant map.

• Subbase. A set of open sets is a subbase (or subbasis) for a topology if every open set in the topology is a union of finite intersections of sets in the subbase. The topology generated by a subbase is the smallest topology containing the subbase elements; this topology consists of all finite intersections of unions of elements of the subbase.

• Subbasis. See Subbase.

• Subcover. A cover K is a subcover (or subcovering) of a cover L if every member of K is a member of L.

• Subcovering. See Subcover.

• Subspace. If X is a space and A is a subset of X, then the subspace topology on A induced by X consists of all intersections of open sets in X with A.

• T₀. A space is T₀ (or Kolmogorov) if for every pair of distinct points x and y in the space, either there is an open set containing x but not y, or there is an open set containing y but not x.

• T₁. A space is T₁ (or accessible) if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T₀; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T₁ if all its singletons are closed. T₁ spaces are always T₀.

• T₂. See Hausdorff.

• T₃. See Regular Hausdorff.

• T_3½. See Tychonoff.

• T₄. See Normal Hausdorff.

• T₅. See Completely normal Hausdorff.

• Topological space. A topological space is a set X equipped with a collection T of subsets of X satisfying the following conditions:

The empty set and X are in T.

The union of any collection of sets in T is also in T.

The intersection of any pair of sets in T is also in T.

The collection T is called a topology on X.

• Topologically complete. A space is topologically complete if it is homeomorphic to a complete metric space.

• Topology. See Topological space.

• Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.

• Trivial topology. The trivial topology on a set X consists of precisely the empty set and the entire space X.

• Tychonoff. A Tychonoff space (or completely regular Hausdorff space, completely T₃ space, T_3½ space) is a completely regular T₀ space. (A completely regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.

• Uniform space. A uniform space is a set U equipped with a nonempty system Φ of subsets of the Cartesian product X × ''X'\' satisfying the following:

if U is in Φ, then U contains { (x, x) : x in X }.

if U is in Φ, then { (y, x) : (x, y) in U } is also in Φ

if U is in Φ and V is a subset of X × X which contains U, then V is in Φ

if U and V are in Φ, then U ∩ V is in Φ

if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U.

The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.

• Uniform structure. See Uniform space.

• Weak topology. The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest topology on the set which makes all the functions continuous.

• Weakly hereditary. A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.

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

Top     

Abbreviations & Acronyms: Open

Top     

Top     

Synonyms within Context: Open

Top     

Top     

Modern Usage: Open

Top     

Commercial Usage: Open

Top     

Image Slideshow: Open

Photos: Open More pictures...

Illustrations: Open More pictures...

Computer Images: Open More pictures...

Top     

Photo Album: Open

Thumbnail	Description & Credit	Thumbnail	Description & Credit
	Long-term cultured AIDS-KS (Kaposi's Sarcoma) cells were injected into a nude mouse. Within 5 days a subcutaneous lesion formed in the induced region with a marked angiogenic response. A cross-section of these new blood vessel formations appear as open spaces, where one can see dividing epithelial cells and some red blood cells. The cells were stained with a Wright-Gimsa stain and are seen at a magnification of 100x. Credit: Unknown photographer/artist.		A typed manuscript lies open on a purple tablecloth. Various types of grain, either loose or in a bowl and a ladle, are on top. There is also a basket with a round loaf of brown bread cut in half. See also AV-3906. Credit: Unknown photographer/artist.
	Histopathology of histoplasmosis in open lung biopsy. FA stain reveals numerous yeast cells of Histoplasma capsulatum. Credit: CDC.		Left ventricle has been cut open to display characteristic severe thickening of mitral valve, thickened chordae tendineae, and hypertrophied left ventricular myocardium. Autopsy. Credit: CDC.
	White Floats out the Open Hatch. Credit: NASA.		Armstrong and Scott with Hatches Open. Credit: NASA.
	Gabby - the talking current buoy Gabby getting a new face for a Norfolk, Virginia, open house MARMER hosted open house for general public Article appeared in Norfolk Ledger-Star on November 13, 1963. Credit: Coast & Geodetic Survey Historical Image Collection.		Boats from HMS EREBUS and HMS TERROR - Captain James Clark Ross Sounded in open ocean at 27.43 S and 17.48 W Recorded depth of approximately 2200 fathoms First modern successful sounding in deep ocean. Credit: Coast & Geodetic Survey Historical Image Collection.
	Natural marsh area adjacent to dredged material deposition area - Open area will be colonized by plants and become productive habitat. Credit: America's Coastlines.		Open pond within mixed vegetation marsh. Credit: America's Coastlines.
Source: pictures compiled by the editor from various references; see picture credits.

Top     

Digital Photo Gallery: Open
 

"America open for business 4" by Vincent Seychal Commentary: "This is a road trip from San Francisco to Albuquerque via NYC between 2001 and 2002."	"Open Hand" by Elisabeth Howe Commentary: "My hand!."
Source: photographs selected by the editor, with permission from the photographers.

Top     

Sounds Captioned with "Open".

Play	Caption	Play	Caption
	The open to Bach's "The Well-Tempered Clavier No. 1".		Open chordal texture with high strings resolving the harmony.
	Pulling open a file cabinet drawer.		Gun going off in the distance in an open field.
	Soda can cracking open.
Source: compiled by the editor from various references; see credits.

Top     

Familiar Quotations: Open

Top     

Historic Usage: Open

Top     

Use in Literature: Open