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Base

Specialty Definition: Base

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Specialty Definition: Base

(From Wikipedia, the free Encyclopedia)

A base is:

• A number that is raised to a power, or base of an exponential function. This finds common use, for example, in the depiction of numbers, for instance, 10 is the base used in the decimal system, whereas 2 is the base in the binary system. See also numeral system and radix
• The base of a logarithmic function.
• One of the parallel sides of a trapezoid or the unequal side of an isosceles triangle.
• Another mathematical meaning is described in the topology glossary.
• Base of a transform in mathematics
• In biochemistry, see nitrogenous base.
• In baseball, a base is one of 4 bags placed at corners of the infield diamond that a player has to run to after hitting the ball.
• In a transistor the base is the controlling connection to the junction.
• BASE jumping is a popular variation on skydiving.
• The name of the terrorist group Al-Qaida translates as "the base."
• In 2001, the catchphrase "All your base are belong to us" swept across the Internet.
• In chemistry, a base is the reactive complement to an acid. (Sometimes the term alkali has been used historically where base is now preferred.) See Acid-base reaction theories. Technically, a substance capable of neutralizing acid, with a pH greater than 7.0. See Table of bases
• Base is also an isolated settlement in inhospitable conditions that must rely on outide help in order to survive, such as military base, Anctarctica base, Moon base.

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

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Base (chemistry)

(From Wikipedia, the free Encyclopedia)

In chemistry, a base is a compound that is the opposite of an acid in the sense that it will neutralize an acid. Common bases include compoundss such as some metal oxides and hydroxides, and ammonia.

An acid "donates" H+ ions to the solution, while a base "accepts" H+ ions _or_ donates OH- ions. Both of those actions will decrease the hydrogen ion (H+) concentration, and thus increase pH (-log[H+])

Soluble bases (alkalis) produce hydroxyl ion (OH^-) in aqueous solution and have a pH above 7.

Example:

The amino group (NH2) acts as a base by accepting a H+ ions from the solution. It does this by forming a coordiate covalent bond with the unshared pair of electrons belonging to the nitrogen atom. This decreases the hydrogen ion concentration.

Sodium hydroxide (NaOH) decomposes into Na+ and OH-, lowering the hydrogen ion concentration because the hydroxide ion will accept hydrogen ions to form water.

See also: acid-base reaction theories. alkaline foods

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

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Numeral system

(From Wikipedia, the free Encyclopedia)

A numeral is a symbol or group of symbols that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article treats the various systems of numerals.

A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the roman numeral for two, the binary numeral for three or the decimal numeral for eleven.

Ideally, a numeration system will represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers), will give every number represented a unique representation (or at least a standard representation) and will reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic.

Numeral systems are sometimes called number systems, but that name is misleading: different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., are not the topic of this article.

Types of numeral systems

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol "|" is chosen, for example, then the number seven would be represented by |||||||. The unary system is normally only useful for small numbers; it has some uses in theoretical computer science.

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if | stands for one, @ for ten and # for 100, then the number 304 can be compactly represented as ### |||| and number 123 as #@@||| . The ancient Egyptian system is of this type, and the Roman system is a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; using our ordinary digits for these abbreviations, we could then write 3# 4| for the number 304. The numeral system of English is of this type ("three hundred four"), as are those of virtually all other languages: Chinese, Japanese, and Greek.

More elegant is a positional system: again working in base 10, we use ten different digits 0,...,9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3*100 + 0*10 + 4. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Arabic numeral system is a positional base 10 system; it is used today throughout the world.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10).

History

Tallies carved from wood and stone have been used since prehistoric times. Stone age cultures, including the American Indians, used tallies for gambling with horses, slaves, personal services and trade-goods.

The earliest known written tallies appear in the ruins of the Sumerian empire, using clay tablets impressed with a sharp stick and baked. The Sumerians had quite an exotic system based on counts to 60, used in astronomical and other calculations. This system was imported to and used by every Mediterranean nation that used astronomy, including the Greeks, Romans and Egyptians. We still use it to count time (minutes per hour), and angle (degrees).

In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in Digital signal processing.

The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman system remained in common use in Europe until positional notation came into common use in the 1500s.

The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.

Some authorities believe that positional arithmetic began with the wide use of the abacus in China. The earliest written positional record seem to be tallies of abacus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932, and seems to have originated as a circle of a place empty of beads.

From China, both the abacus and written tallies may have moved to India, perhaps via Chinese traders and businesses. In India, recognizably modern positional numeral systems, used for astronomy and accounting, appeared in the Mogul empire.

From India, the thriving trade between Islamic Moguls and Africa carried the concept to Cairo. Arabic mathematicians extended the system to decimal fractions, and al-Khwarizmi wrote an important work about it in the 9th century. The system was introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisas Liber Abaci of 1201.

The binary system (base 2), propagated in the 17th century by Gottfried Leibniz who had heard about it from China, came in common use in the 20th century because of computer applications.

Bases used

The base-10 system, the one most commonly used by humans today, originated because we have ten fingers, thus allowing for simple counting. A base-eight system was devised by (at least) the Yuki Pomo of Northern California, who used the spaces between the fingers to count. The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base 20, (possibly originating from the number of a person's fingers and toes). Base 60 was used by the Sumerians and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds). Base-12 systems were popular because multiplication is easier in them than in base-10, (addition just as easy) and because the year has twelve months; we still have a special word for "dozen" and use 12 hours for every night and day.

Electronic components (first vacuum tubes, then transistors) may have only 2 possible states: concat(1) and closed (0). Because this is exactly the set of binary digits, and because arithmetics in a binary system are the easiest to describe electronically (using Boolean algebra), the binary system became natural for electronic computers. It is used to perform integer arithmetic in almost all electronic computers (the only exception being the exotic base-3 and base-10 designs that were discarded very early in the history of computing hardware). Note however that a computer does not treat all of its data as integers. Thus, some of it may be treated as texts and program data. Real numbers (numbers that can be not whole) are usually written down in the floating point notation, that has different rules of arithmetic.

Positional systems in detail

In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b.

For example, in the decimal system (base 10), the numeral 4327 means (4×10³) + (3×10²) + (2×10¹) + (7×10⁰), noting that 10⁰ = 1.

In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a₁b^k + a₂b^k-1 + a₃b^k-2 + ... + a_k+1b⁰ and writing the digits a₁a₂a₃ ... a_k+1 in order. The digits are natural numbers between 0 and b-1, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base is added in subscript to the right of the number, like this: number_base. Numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×2¹+ 0×2⁰ +1×2^-1 +1×2^-2 = 2.75.

Note that a number has a terminating or repeating expansions if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.3₁₀ = 0.0100110011001...₂). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all bases. Thus, for example in base 2, &pi = 3.1415926...₁₀ can be written down as the unperiodic 11.001001000011111...₂.

If b=p is a prime number, one can define base-p numerals whose expansion to the left never stops; these are called the p-adic numbers.

Specific numeral systems

Positional systems

• b=2: Binary
• b=3: Ternary
• b\=4: Quaternary
• b=5: Quinary
• b=6: Senary
• b=7: Septimal
• b=8: Octal or Octonary
• b=9: Novenary
• b=10: Decimal or Denary (see Arabic numerals, Armenian numerals, Chinese numerals, Greek numerals, Hebrew numerals, Indian numerals, Japanese numerals, Roman numerals)
• b=12: Duodecimal (used in the Chepang language of Nepal)
• b=16: Hexadecimal
• b=20: Vigesimal (see Mayan numerals)
• b=27: Base 27
• b=60: Sexagesimal (see Babylonian numerals)

Positional-like systems with non-standard bases

• b=-3: Negaternary
• b=-2: Negabinary
• b=&phi: Golden mean base
• b=2i: Quater-imaginary base
• b=sqrt(2)i: Binary square-root-2 times i base
• b=i - 1: Binary i-1 base
• b=Fibonacci number: Fibonacci coding
• b varies: Mixed radix

Other systems

• Unary numeral system
• Roman numerals

See also: Computer numbering formats

External Resources

D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. Addison-Wesley. pp.194-213, "Positional Number Systems"

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

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PH

(From Wikipedia, the free Encyclopedia)

Alternate uses: see PH (disambiguation)

pH is a measure of the concentration of protons (H⁺) in a solution and, therefore, its acidity or alkalinity. The concept was introduced by S.P.L. Sørensen in 1909. The p stands for the German potenz, meaning power or concentration, and the H for the hydrogen ion (H⁺).

The formula for calculating pH is:

where [H⁺] indicates the concentration of H⁺ ions (or also written [H₃O⁺], concentration of the equivalent hydronium ions), measured in moles per litre (also known as molarity). Or in layman's terms , the " pH " value is an approximate number between 0 and 14 for the negative decimal log of the hydronium - ion concentration .

In aqueous solution at standard temperature and pressure, a pH of 7 indicates neutrality (e.g. pure water) because Water naturally disassociates into H⁺ and OH^- ions with equal concentrations of 1×10^-7M. A lower pH number (for example pH 3) indicates increasing strength of acidity, and a higher pH number (for example pH 11) indicates increasing strength of alkalinity. Most substances have a pH in the range 0 to 14, although extremely acidic or basic substances may have pH < 0, or pH > 14.

In nonaqueous solutions or non-STP conditions, the pH of neutrality may not be 7. Instead it is related to the disassociation constant for the specific solvent used.

There is also pOH, in a sense the opposite of pH, which measures the concentration of OH^- ions. Since water self ionizes, and notating [OH^-] as the concentration of hydroxide ions, we have

K_w=[H⁺][OH^-]=10^-14

where K_w is constant, the ionization constant of water.

Now, since

log K_w=log [H⁺] + log [OH^-]

by logarithmic identities, we then have the relationship

14 = log [H⁺] + log [OH^-]

and thus

pOH = log [OH^-] = 14 - log [H⁺]

Some common aqueous pH's

• 3.5: orange juice (slightly acidic)
• 5.6: unpolluted rain water (slightly acidic)
• 7.0: pure water
• 7.34 - 7.45: human blood (slightly alkaline)
• 11.0: household ammonia (very alkaline)

Measuring

pH can be measured by addition of a pH indicator or using a pH meter. Universal Indicator changes colour depending on the pH of the solution it is added to. Electronic pH meters consist of an electrolytic cell in which an electric current is created due to the hydrogen cations completing the circuit.

Calculation of pH for weak and strong acids

Values of pH for weak and strong acids can be approximated using certain assumptions. It is assumed that for strong acids, the dissociation reaction goes to completion (i.e., no unreacted acid remains in solution). Dissolving the strong acid HCl in water can therefore be expressed:

HCl(aq) → H⁺ + Cl^-

This means that in a 0.01 M solution of HCl it is approximated that there is a concentration of 0.01 M dissolved hydrogen ions. From above, the pH is: pH = -log₁₀ [H⁺(aq)]:

pH = -log(0.01)

which equals 2.

For weak acids the dissociation reaction does not go to completion, an equlibrium is set up between the ions and the acid. The following shows the equilibrium reaction between methanoic acid and its ions:

HCOOH(aq) ↔ H⁺(aq) + HCOO^-(aq)

It is necessary to know the value of the equilibrium constant of the reaction for each acid in order to calculate its pH. In the context of pH, this is termed the acidity constant of the acid but is worked out in the same way (see chemical equilibrium):

K_a = [hydrogen ions (aq)][acid ions (aq)] / [acid (aq)]

For HCOOH, K_a = 1.6 × 10^-4

Two assumptions are made in the calculation of pH for a weak acid. It is assumed that the water the acid is dissolved in does not provide any hydrogen ions. Water is a very weak acid and in general it supplies far fewer than the acid dissolved in it. Consequently in the above reaction the concentration of hydrogen ions equals the concentration of methanoate ions:

[H^-(aq)] = [HCOO^-(aq)]

It is also taken that the amount of undissociated acid at equilibrium is equal to the amount originally added to the solution. Although this is obviously untrue (otherwise the pH would remain 7!) this amount can be neglected because the fraction of hydrogen ions given is again very small.

With a 0.1 M solution of methanoic acid (HCOOH), the acidity constant is equal to:

K_a = [H⁺(aq)][HCOO^-(aq)] / [HCOOH(aq)]

So:

1.6 × 10^-4 = [H⁺][HCOO^-] / 0.1

1.6 × 10^-4 × 0.1 =[H⁺][HCOO^-]

As [H^-(aq)] = [HCOO^-(aq)]:

1.6 × 10^-4 × 0.1 =[H⁺]²

The concentration of hydrogen ions is: 4 × 10^-3. The pH, therefore, is: 2.3.

See also: Acid-base reaction theories, Acid, Base, Alkali, Soil pH, titration

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

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Ph

(From Wikipedia, the free Encyclopedia)

ph or PH may be:

• pH value
• Philippines (ISO country code)
• phot (ph) unit

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

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Radix

(From Wikipedia, the free Encyclopedia)

Radix (from Latin, basis) is the number base of a numeral system. e.g., binary is "base 2" and has a radix of 2. When describing radix in mathematical notation, b is generally used as a symbol for this concept; so, for a binary system, b equals 2.

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

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Root system

(From Wikipedia, the free Encyclopedia)

See also Simple Lie group.

In mathematics, a root system is a kind of configuration in Euclidean space that has turned out to be fundamental in Lie group theory. Since Lie groups (and some analogues such as algebraic groups) became used in most parts of mathematics during the twentieth century, the apparently special nature of of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie groups (such as singularity theory.

Definitions

Formally, a root system is a finite set Φ of non-zero vectors (roots) spanning a finite-dimensional Euclidean space V which satisfy the following properties:

1. The only scalar multiples of a root α in V which belong to Φ are α itself and −α.
2. For every root α in V, the set Φ is symmetric under reflection through the hyperplane of vectors perpendicular to α
3. If α and β are vectors in Φ, the projection of 2β onto the line through α is an integer multiple of α

The rank of a root system Φ is the dimension of V. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A₂, B₂, and G₂ pictured below, is said to be irreducible.

Two irreducible root systems (E₁,Φ₁) and (E₂,Φ₂) are considered to be the same if there is an invertible linear transformation E₁→E₂ which preserves distance up to a scale factor and which sends Φ₁ to Φ₂.

The group of isometries of V generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ as it acts faithfully on the finite set Φ, the Weyl group is always finite.

Classification

It is not too difficult to classify the root systems of rank 2:

Whenever Φ is a root system in V and W is a subspace of V spanned by Ψ=Φ∩W, then Ψ is a root system in W. Thus, our exhaustive list of root systems of rank 2 shows the geometric possibilities for any two roots in a root system. In particular, two such roots meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

In general, irreducible root systems are specified by a family (indicated by a letter A to G) and the rank (indicated by a subscript). There are four infinite families and five exceptional cases:

• A_n (n≥1)
• B_n (n≥2)
• C_n (n≥3)
• D_n (n≥4)
• E₆
• E₇
• E₈
• F₄
• G₂

Dynkin diagrams

To prove this classification theorem, one uses the angles between pairs of roots to encode the root system in a much simpler combinatorial object, the Dynkin diagram. The Dynkin diagrams can then be classified according to the scheme given above.

To every root system is associated a graph (possibly with a specially marked edge) called the Dynkin diagram. The Dynkin diagram can be extracted from the root system by choosing a base, that is a subset Δ of Φ which is a basis of V with the special property that every vector in Φ when written in the basis Δ has either all coefficients ≥0 or else all ≤0.

The vertices of the Dynkin diagram correspond to vectors in Δ. An edge is drawn between each non-orthogonal pair of vectors; it is a double edge if they make an angle of 135 degrees, and a triple edge if they make an angle of 150 degrees. In addition, double and triple edges are marked with an angle sign pointing toward the shorter vector.

Although a given root system has more than one base, the Weyl group acts transitively on the set of bases. Therefore, the root system determines the Dynkin diagram. Given two root systems with the same Dynkin diagram, we can match up roots, starting with the roots in the base, and show that the systems are in fact the same.

Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams, and the problem of classifying irreducible root systems reduces to the problem of classifying connected Dynkin diagrams. Dynkin diagrams encode the inner product on E in terms of the basis Δ, and the condition that this inner product must be positive definite turns out to be all that is needed to get the desired classification. The actual connected diagrams are as follows:

In detail, the individual root systems can be realized case-by-case, as in the following sections.

A_n

Let V be the subspace of Rⁿ⁺¹ for which the coordinates sum to 0, and let Φ be the set of vectors in V of length √2 and with integer coordinates in Rⁿ⁺¹. Such a vector must have all but two coordinates equal to 0, one coordinate equal to 1, and one equal to -1, so there are n²+n roots in all.

B_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of length 1 or √2. The total number of roots is 2n².

C_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of √2 together with all vectors of the form 2λ, where λ is an integer vector of length 1. The total number of roots is 2n². The total number of roots is 2n².

D_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of length √2. The total number of roots is 2n².

E_n

For V₈, let V=R⁸, and let E₈ denote the set of vectors α of length √2 such that the coordinates of 2α are all integers and are either all even or all odd. Then E₇ can be constructed as the intersection of E₈ with the hyperplane of vectors perpendicular to a fixed root α in E₈, and E₆ can be constructed as the intersection of E₈ with two such hyperplanes corresponding to roots α and β which are neither orthogonal to one another nor scalar multiples of one another. The root systems E₆, E₇, and E₈ have 72, 126, and 240 roots respectively.

F₄

For F₄, let V=R⁴, and let Φ denote the set of vectors α of length 1 or √2 such that the coordinates of 2α are all integers and are either all even or all odd. There are 48 roots in this system.

G₂

There are 12 roots in G₂, which form the vertices of a hexagram. See the picture above.

Root systems and Lie theory

Irreducible root systems classify a number of related objects in Lie theory, notably:

• Simple complex Lie algebras
• Simple complex Lie groups
• Simply connected complex Lie groups which are simple modulo their centers
• Simple compact Lie groups

In each case, the roots are non-zero weightss of the adjoint representation.

See also Weyl group, Coxeter group, Cartan matrix, Coxeter matrix

A root system can also be said to describe a plant's roots and associated systems.

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

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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."

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Abbreviations & Acronyms: Base

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Photo Album: Base

Thumbnail	Description & Credit	Thumbnail	Description & Credit
	If there is a pair of tufts near the base of the siphon, and a ventral row of tufts, or a row of straight hairs following the pecten, the genus is identified as Culiseta. Credit: CDC.		Science Base on Lunar Farside. Credit: NASA.
	Tranquility Base. Credit: NASA.		Aldrin Looks Back at Tranquility Base. Credit: NASA.
	White River West Base on the 141st Meridian Survey International Boundary Commission surveyors. Credit: Coast & Geodetic Survey Historical Image Collection.		100-foot tower at Bozeman NW Base Note man at top Triangulation party of William M. Scaife. Credit: Coast & Geodetic Survey Historical Image Collection.
	Minnows that constitute the forage base for larger predators. Credit: America's Coastlines.		A decorative lighthouse graces the entrance to the United States Coast Guard Base at Charleston. Credit: America's Coastlines.
	Tigvariak Island base camp from the air. Credit: Paths Less Taken - NOAA at the Ends of the Earth.		The abandoned base at Port Lockroy. The SHACKLETON came here to remove equipment. 64 50 S Latitude 63 30 W Longitude. Credit: Paths Less Taken - NOAA at the Ends of the Earth.
Source: pictures compiled by the editor from various references; see picture credits.

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Digital Photo Gallery: Base
 

"Candle and brass base" by