User:Ascii/Definitions (by Meaning 1-800)

Metaconcepts

 * Definition:Mathematics
 * Definition:Natural Language


 * Definition:Object
 * Definition:Symbol
 * Definition:Property
 * Definition:Constant
 * Definition:Variable
 * Definition:Unique
 * Definition:Distinct
 * Definition:Distinct/Singular
 * Definition:Distinct/Plural
 * Definition:Operation


 * Definition:Formal Language
 * Definition:Formal Language/Alphabet
 * Definition:String
 * Definition:Expression
 * Definition:Well-Formed Formula
 * Definition:Formal Semantics


 * Definition:Algorithm
 * Definition:Primitive Recursive/Function
 * Definition:Unlimited Register Machine

Logic

 * Definition:Logic
 * Logic is the study of the structure of statements and their truth values, divorced from their conceptual content.
 * Definition:Symbolic Logic
 * Symbolic logic is the study of logic in which the logical form of statements is analyzed by using symbols as tools. Instead of explicit statements, logical formulas are investigated, which are symbolic representations of statements, and compound statements in particular. In symbolic logic, the rules of reasoning and logic are investigated by means of formal systems, which form a good foundation for the symbolic manipulations performed in this field.


 * Definition:Statement
 * A statement is a sentence which has objective and logical meaning.
 * Definition:Proposition
 * A proposition is a statement which is offered up for investigation as to its truth or falsehood. Loosely, a proposition is a statement which is about to be proved (or disproved).
 * Definition:True
 * A statement has a truth value of true what it says matches the way that things are.
 * Definition:False
 * A statement has a truth value of false what it says does not match the way that things are.
 * Definition:Truth Value
 * In Aristotelian logic, a statement can be either true or false, and there is no undefined, in-between value. Whether it is true or false is called its truth value. Note that a statement's truth value may change depending on circumstances.
 * Definition:Proof System
 * Let $\mathcal L$ be a formal language. A proof system $\mathscr P$ for $\mathcal L$ comprises:
 * Axioms and/or axiom schemata;
 * Rules of inference for deriving theorems.
 * It is usual that a proof system does this by declaring certain arguments concerning $\mathcal L$ to be valid. Informally, a proof system amounts to a precise account of what constitutes a (formal) proof.
 * Definition:Axiom
 * In all contexts, the definition of the term axiom is by and large the same. That is, an axiom is a statement which is accepted as being true. A statement that is considered an axiom can be described as being axiomatic.
 * Definition:Assumption
 * An assumption is a statement which is introduced into an argument, whose truth value is (temporarily) accepted as True. In mathematics, the keyword let is often the indicator here that an assumption is going to be introduced. For example: Let $p$ (be true) ... can be interpreted, in natural language, as: Let us assume, for the sake of argument, that $p$ is true ...
 * Definition:Premise
 * A premise is an assumption that is used as a basis from which to start to construct an argument. When the validity or otherwise of a proof is called into question, one may request the arguer to "check your premises".


 * Definition:Valid Argument
 * A valid argument is a logical argument in which the premises provide conclusive reasons for the conclusion.
 * Definition:Sequent
 * A sequent is an expression in the form: $\phi_1, \phi_2, \ldots, \phi_n \vdash \psi$ where $\phi_1, \phi_2, \ldots, \phi_n$ are premises (any number of them), and $\psi$ the conclusion (only one), of an argument.
 * Definition:Logical Language
 * Let $\mathcal L$ be a formal language used in symbolic logic. Then $\mathcal L$ is called a logical language.
 * Definition:Logical Formula
 * Let $\mathcal L$ be a formal language used in the field of symbolic logic. Then the well-formed formulas of $\mathcal L$ are often referred to as logical formulas. They are symbolic representations of statements, and often of compound statements in particular.
 * Definition:Parenthesis
 * Parenthesis is a syntactical technique to disambiguate the meaning of a logical formula. It allows one to specify that a logical formula should (temporarily) be regarded as being a single entity, being on the same level as a statement variable. Such a formula is referred to as being in parenthesis. Typically, a formal language, in defining its formal grammar, ensures by means of parenthesis that all of its well-formed words are uniquely readable. Generally, brackets are used to indicate that certain formulas are in parenthesis. The brackets that are mostly used are round ones, the left (round) bracket $($ and the right (round) bracket $)$.


 * Definition:Propositional Logic
 * Propositional logic is a sub-branch of symbolic logic in which the truth values of propositional formulas are investigated and analysed. The atoms of propositional logic are simple statements. There are various systems of propositional logic for determining the truth values of propositional formulas, for example:
 * Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either "self-evident" axioms or themselves derived from other valid sequents.
 * The Method of Truth Tables, which consists of the construction of one or more truth tables which exhaustively list all the possible truth values of all the statement variables with a view to determining the required answer by inspection.
 * Definition:Language of Propositional Logic
 * There are a lot of different formal systems expressing propositional logic. Although they vary wildly in complexity and even disagree (to some extent) on what expressions are valid, generally all of these use a compatible formal language.


 * Definition:Logical Connective
 * Definition:Logical Not
 * Definition:Conjunction
 * Definition:Disjunction
 * Definition:Conditional
 * Definition:Conditional/Necessary Condition
 * Definition:Converse Statement
 * Definition:Biconditional
 * Definition:Logical Equivalence
 * Definition:Iff
 * Definition:Vacuous Truth
 * Definition:Main Connective/Propositional Logic
 * Definition:Propositional Function
 * Definition:Boolean Interpretation
 * Definition:Tautology
 * Definition:Contradiction
 * Definition:Language of Propositional Logic/Formal Grammar/WFF


 * Definition:Categorical Statement
 * Definition:Categorical Syllogism


 * Definition:Predicate Logic
 * Definition:Universal Quantifier
 * Definition:Existential Quantifier
 * Definition:Free Variable
 * Definition:Language of Predicate Logic
 * Definition:Language of Predicate Logic/Formal Grammar
 * Definition:Structure for Predicate Logic


 * Definition:By Hypothesis
 * Definition:Mutatis Mutandis
 * Definition:WLOG


 * Definition:Basis for the Induction
 * Definition:Induction Hypothesis
 * Definition:Induction Step


 * Definition:RHS

Set Theory

 * Definition:Set Theory


 * Definition:Set
 * Definition:Element
 * Definition:Subset
 * Definition:Proper Subset
 * Definition:Subset/Superset
 * Definition:Set Equality
 * Definition:Singleton
 * Definition:Doubleton
 * Definition:Empty Set
 * Definition:Non-Empty Set
 * Definition:Disjoint Sets
 * Definition:Pairwise Disjoint
 * Definition:Set of Sets
 * Definition:Power Set
 * Definition:Set Partition
 * Definition:Universe (Set Theory)


 * Definition:Set Union
 * Definition:Set Union/Set of Sets
 * Definition:Set Union/Countable Union
 * Definition:Set Union/Family of Sets
 * Definition:Set Intersection
 * Definition:Set Difference
 * Definition:Symmetric Difference
 * Definition:Set Complement
 * Definition:Relative Complement
 * Definition:Successor Set


 * Definition:Finite
 * Definition:Finite Set
 * Definition:Countable Set
 * Definition:Countable Set/Countably Infinite
 * Definition:Infinite
 * Definition:Infinite Set
 * Definition:Uncountable Set
 * Definition:Set Equivalence
 * Definition:Cardinality


 * Definition:Indexing Set
 * Definition:Indexing Set/Family
 * Definition:Indexing Set/Family of Sets
 * Definition:Indexing Set/Family of Subsets
 * Definition:Indexing Set/Indexed Set


 * Definition:Ordered Pair
 * Definition:Cartesian Product
 * Definition:Cartesian Product/Finite
 * Definition:Relation
 * Definition:Relational Structure
 * Definition:Domain (Set Theory)/Relation
 * Definition:Codomain (Set Theory)/Relation
 * Definition:Image (Set Theory)/Relation/Subset
 * Definition:Restriction/Relation
 * Definition:Inverse Relation
 * Definition:Endorelation
 * Definition:Reflexive Relation
 * Definition:Antireflexive Relation
 * Definition:Reflexivity
 * Definition:Symmetric Relation
 * Definition:Antisymmetric Relation
 * Definition:Asymmetric Relation
 * Definition:Transitive Relation
 * Definition:Connected Relation
 * Definition:One-to-Many Relation
 * Definition:Many-to-One Relation
 * Definition:Left-Total Relation
 * Definition:Left-Total Relation/Multifunction
 * Definition:Composition of Relations


 * Definition:Diagonal Relation
 * Definition:Equivalence Relation
 * Definition:Equivalence Class
 * Definition:Quotient Set


 * Definition:Ordered Tuple
 * Definition:Ordered Tuple as Ordered Set/Ordered Triple


 * Definition:Preordering/Preordered Set
 * Definition:Directed Subset


 * Definition:Ordering
 * Definition:Ordered Set


 * Definition:Strictly Precede


 * Definition:Upper Closure/Element
 * Definition:Lower Closure/Element
 * Definition:Interval/Ordered Set/Closed


 * Definition:Upper Bound of Set
 * Definition:Lower Bound of Set
 * Definition:Supremum of Set
 * Definition:Infimum of Set
 * Definition:Up-Complete


 * Definition:Closure Operator
 * Definition:Continuous Ordered Set


 * Definition:Maximal/Ordered Set
 * Definition:Minimal/Ordered Set
 * Definition:Greatest/Ordered Set
 * Definition:Smallest/Ordered Set
 * Definition:Smallest Set by Set Inclusion
 * Definition:Bounded Above Set
 * Definition:Bounded Below Set


 * Definition:Upper Set
 * Definition:Lower Set


 * Definition:Ideal in Ordered Set
 * Definition:Ideal (Order Theory)


 * Definition:Element is Way Below
 * Definition:Compact Element


 * Definition:Total Ordering
 * Definition:Totally Ordered Set
 * Definition:Chain (Set Theory)
 * Definition:Max Operation


 * Definition:Well-Ordering
 * Definition:Strict Well-Ordering
 * Definition:Well-Ordered Set


 * Definition:Lattice
 * Definition:Join Semilattice
 * Definition:Meet Semilattice
 * Definition:Complete Lattice


 * Definition:Minimal Infinite Successor Set


 * Definition:Class (Class Theory)
 * Definition:Class/Proper Class
 * Definition:Transitive Class
 * Definition:Small Class


 * Definition:Cardinal


 * Definition:Ordinal Class
 * Definition:Limit Ordinal


 * Definition:Ordered Subset

Mappings

 * Definition:Mapping
 * Definition:Graph of Mapping
 * Definition:Well-Defined/Mapping
 * Definition:Function
 * Definition:Domain (Set Theory)/Mapping
 * Definition:Codomain (Set Theory)/Mapping
 * Definition:Image (Set Theory)/Mapping/Mapping
 * Definition:Image (Set Theory)/Mapping/Subset
 * Definition:Image (Set Theory)/Mapping/Element
 * Definition:Preimage/Mapping/Subset
 * Definition:Preimage/Mapping/Element
 * Definition:Fixed Point
 * Definition:Restriction/Mapping
 * Definition:Extension of Mapping
 * Definition:Inverse Mapping
 * Definition:Finite Sequence
 * Definition:Constant Mapping
 * Definition:Identity Mapping
 * Definition:Permutation
 * Definition:Surjection
 * Definition:Injection
 * Definition:Bijection


 * Definition:Projection (Mapping Theory)
 * Definition:Projection (Mapping Theory)/First Projection
 * Definition:Direct Image Mapping/Mapping


 * Definition:Set of All Mappings
 * Definition:Composition of Mappings


 * Definition:Characteristic Function (Set Theory)/Set
 * Definition:Inclusion Mapping


 * Definition:Sequence
 * Definition:Length of Sequence
 * Definition:Sequence/Infinite Sequence
 * Definition:Term of Sequence
 * Definition:Subsequence


 * Definition:Bounded Mapping
 * Definition:Bounded Sequence/Real
 * Definition:Bounded Below


 * Definition:Order Isomorphism
 * Definition:Order Embedding
 * Definition:Increasing/Mapping
 * Definition:Quotient Mapping


 * Definition:Ordinal

Number Systems

 * Definition:Number
 * Definition:Natural Numbers
 * Definition:Zero (Number)
 * Definition:Number Base
 * Definition:Decimal Notation
 * Definition:Digit
 * Definition:Decimal Expansion
 * Definition:Initial Segment
 * Definition:Initial Segment of Natural Numbers


 * Definition:Integer
 * Definition:Positive/Integer
 * Definition:Strictly Positive/Integer
 * Definition:Negative/Integer
 * Definition:Rational Number
 * Definition:Rational Number/Fraction
 * Definition:Rational Number/Fraction/Numerator
 * Definition:Rational Number/Fraction/Denominator
 * Definition:Rational Number/Canonical Form
 * Definition:Ratio


 * Definition:Medial
 * Definition:Irrational Number
 * Definition:Commensurable
 * Definition:Incommensurable
 * Definition:Commensurable in Square Only
 * Definition:Incommensurable in Square


 * Definition:Real Number
 * Definition:Real Number/Axioms
 * Definition:Positive/Real Number
 * Definition:Strictly Positive
 * Definition:Strictly Positive/Real Number
 * Definition:Negative/Real Number
 * Definition:Real Number/Real Number Line
 * Definition:Real Interval
 * Definition:Real Interval/Open
 * Definition:Real Interval/Closed
 * Definition:Real Interval/Half-Open
 * Definition:Coordinate System/Origin
 * Definition:Cartesian Coordinate System
 * Definition:Cartesian Plane
 * Definition:Axis/X-Axis
 * Definition:Axis/Y-Axis


 * Definition:Complex Number
 * Definition:Complex Number/Real Part
 * Definition:Complex Number/Imaginary Part
 * Definition:Complex Number/Imaginary Unit
 * Definition:Complex Number/Polar Form
 * Definition:Argument of Complex Number
 * Definition:Complex Conjugate
 * Definition:Complex Plane
 * Definition:Complex Arithmetic


 * Definition:Quaternion


 * Definition:Absolute Value
 * Definition:Addition
 * Definition:Addition/Natural Numbers
 * Definition:Addition/Integers
 * Definition:Addition/Real Numbers
 * Definition:Complex Addition
 * Definition:Multiplication
 * Definition:Multiplication/Natural Numbers
 * Definition:Multiplication/Integers
 * Definition:Multiplication/Real Numbers
 * Definition:Complex Multiplication
 * Definition:Power (Algebra)/Integer
 * Definition:Power (Algebra)/Real Number
 * Definition:Complex Modulus


 * Definition:Standard Number Field


 * Definition:Usual Ordering

Number Theory

 * Definition:Even Integer
 * Definition:Odd Integer
 * Definition:Reciprocal
 * Definition:Proportion


 * Definition:Floor Function
 * Definition:Ceiling Function


 * Definition:Square Number
 * Definition:Square/Function
 * Definition:Square Root
 * Definition:Quadratic Equation


 * Definition:Factorial
 * Definition:Falling Factorial
 * Definition:Integer Sequence
 * Definition:Divisor (Algebra)/Integer
 * Definition:Divisor (Algebra)/Integer/Aliquot Part
 * Definition:Common Divisor of Integers
 * Definition:Greatest Common Divisor/Integers
 * Definition:Lowest Common Multiple/Integers
 * Definition:Integers Modulo m
 * Definition:Modulo Operation
 * Definition:Congruence (Number Theory)
 * Definition:Congruence Modulo Integer
 * Definition:Quadratic Residue


 * Definition:Prime Number
 * Definition:Composite Number
 * Definition:Odd Prime
 * Definition:Semiprime Number
 * Definition:Prime Decomposition
 * Definition:Prime Decomposition/Multiplicity
 * Definition:Square-Free Integer
 * Definition:Prime Factor
 * Definition:Coprime Integers
 * Definition:Perfect Number


 * Definition:Fibonacci Number
 * Definition:Golden Mean


 * Definition:Apotome


 * Definition:Sigma Function
 * Definition:Tau Function
 * Definition:Euler Phi Function
 * Definition:Deficient Number


 * Definition:Triangular Number
 * Definition:Pascal's Triangle
 * Definition:Binomial (Euclidean)
 * Definition:Binomial Coefficient


 * Definition:Stirling Numbers of the First Kind/Unsigned
 * Definition:Stirling Numbers of the Second Kind


 * Definition:Geometric Sequence
 * Definition:Geometric Sequence/Integers


 * Definition:Real Sequence
 * Definition:Complex Sequence
 * Definition:Complex Roots of Unity
 * Definition:Complex Function


 * Definition:Algebraic Number
 * Definition:Harmonic Numbers
 * Definition:P-adic Norm
 * Definition:P-adic Number


 * Definition:Convergent of Continued Fraction

Abstract Algebra

 * Definition:Abstract Algebra


 * Definition:Well-Defined/Operation
 * Definition:Operation/Binary Operation
 * Definition:Closure (Abstract Algebra)/Algebraic Structure
 * Definition:Product Notation (Algebra)/Infinite
 * Definition:Restriction/Operation
 * Definition:Pointwise Operation


 * Definition:Subset Product
 * Definition:Idempotence/Element
 * Definition:Idempotence/Operation
 * Definition:Commute
 * Definition:Commutative Operation
 * Definition:Associative Operation
 * Definition:Cancellable Element
 * Definition:Identity (Abstract Algebra)/Two-Sided Identity
 * Definition:Invertible Element
 * Definition:Inverse (Abstract Algebra)/Inverse


 * Definition:Relation Compatible with Operation
 * Definition:Congruence Relation


 * Definition:Algebraic Structure
 * Definition:Underlying Set
 * Definition:Order of Structure
 * Definition:Morphism Property
 * Definition:Homomorphism (Abstract Algebra)
 * Definition:Epimorphism (Abstract Algebra)
 * Definition:Isomorphism (Abstract Algebra)
 * Definition:External Direct Product


 * Definition:Magma
 * Definition:Semigroup
 * Definition:Subsemigroup
 * Definition:Naturally Ordered Semigroup
 * Definition:Commutative Semigroup
 * Definition:Monoid
 * Definition:Commutative Monoid
 * Definition:Group
 * Definition:Trivial Group
 * Definition:Additive Group of Integers
 * Definition:Group Axioms
 * Definition:Group Product
 * Definition:Group Direct Product
 * Definition:Power of Element/Group
 * Definition:Finite Group
 * Definition:Cayley Table
 * Definition:Order of Group Element
 * Definition:Abelian Group
 * Definition:Subgroup
 * Definition:Index of Subgroup
 * Definition:Normal Subgroup
 * Definition:Generated Subgroup
 * Definition:Generator of Subgroup
 * Definition:Sylow p-Subgroup
 * Definition:Cyclic Group
 * Definition:Cyclic Group/Generator
 * Definition:Center (Abstract Algebra)/Group
 * Definition:Symmetric Group
 * Definition:Symmetric Group/n Letters
 * Definition:Normalizer


 * Definition:Group Homomorphism
 * Definition:Kernel of Group Homomorphism
 * Definition:Isomorphism (Abstract Algebra)/Group Isomorphism
 * Definition:Group Epimorphism
 * Definition:Coset/Left Coset
 * Definition:Coset/Right Coset
 * Definition:Quotient Group
 * Definition:Orbit (Group Theory)


 * Definition:Distributive Operation
 * Definition:Boolean Algebra


 * Definition:Ring (Abstract Algebra)
 * Definition:Ring (Abstract Algebra)/Ring Axioms
 * Definition:Ring (Abstract Algebra)/Addition
 * Definition:Ring (Abstract Algebra)/Product
 * Definition:Kronecker Delta
 * Definition:Subring
 * Definition:Ring Zero
 * Definition:Commutative Ring
 * Definition:Unity (Abstract Algebra)/Ring
 * Definition:Ring with Unity
 * Definition:Divisor (Algebra)/Ring with Unity
 * Definition:Commutative and Unitary Ring
 * Definition:Unit of Ring
 * Definition:Group of Units/Ring
 * Definition:Ideal of Ring
 * Definition:Principal Ideal of Ring
 * Definition:Maximal Ideal of Ring
 * Definition:Division Ring
 * Definition:Norm/Division Ring
 * Definition:Normed Division Ring
 * Definition:Metric Induced by Norm on Division Ring
 * Definition:Non-Archimedean/Norm (Division Ring)
 * Definition:Convergent Sequence/Normed Division Ring
 * Definition:Cauchy Sequence/Normed Division Ring
 * Definition:Quotient Ring
 * Definition:Ring Homomorphism
 * Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism
 * Definition:Ring Monomorphism
 * Definition:Characteristic of Ring
 * Definition:Algebra over Ring


 * Definition:Integral Domain
 * Definition:Strict Positivity Property
 * Definition:Proper Zero Divisor
 * Definition:Ordered Basis
 * Definition:Ordered Integral Domain
 * Definition:Linearly Independent/Set


 * Definition:Field (Abstract Algebra)
 * Definition:Subfield
 * Definition:Field Extension
 * Definition:Field of Quotients
 * Definition:Valued Field
 * Definition:Field of Real Numbers


 * Definition:Summation
 * Definition:Summation/Indexed


 * Definition:Polynomial
 * Definition:Polynomial over Ring
 * Definition:Polynomial Ring
 * Definition:Polynomial Ring/Indeterminate
 * Definition:Coefficient of Polynomial
 * Definition:Ring of Polynomial Forms


 * Definition:Module
 * Definition:Right Module
 * Definition:Unitary Module
 * Definition:Linear Transformation
 * Definition:Dimension (Linear Algebra)
 * Definition:Generator of Module


 * Definition:Vector Space
 * Definition:Vector (Linear Algebra)
 * Definition:Zero Vector
 * Definition:Vector Subspace
 * Definition:Real Vector Space
 * Definition:Dimension of Vector Space
 * Definition:Normed Vector Space
 * Definition:Norm/Vector Space
 * Definition:Basis of Vector Space
 * Definition:Vector-Valued Function
 * Definition:Linear Combination
 * Definition:Standard Ordered Basis/Vector Space


 * Definition:Coefficient
 * Definition:Degree of Polynomial
 * Definition:Root of Polynomial
 * Definition:Group Action


 * Definition:Matrix
 * Definition:Matrix/Element
 * Definition:Matrix/Row
 * Definition:Matrix/Column
 * Definition:Matrix/Square Matrix
 * Definition:Matrix/Square Matrix/Order
 * Definition:Matrix Space
 * Definition:Unit Matrix
 * Definition:Matrix Product (Conventional)
 * Definition:Determinant/Matrix

Metric Spaces

 * Definition:Pseudometric
 * A pseudometric on a set $A$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions:
 * $(M1) : \forall x \in A : d \left({x, x}\right) = 0$
 * $(M2) : \forall x, y, z \in A : d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$
 * $(M3) : \forall x, y \in A : d \left({x, y}\right) = d \left({y, x}\right)$
 * Definition:Pseudometric/Pseudometric Space
 * A pseudometric space $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.
 * Definition:Metric Space
 * Definition:Metric Space Axioms
 * Definition:Metric Space/Distance Function
 * Definition:Metric Subspace


 * Definition:Euclidean Metric/Real Number Line
 * Definition:Isometry (Metric Spaces)


 * Definition:Open Ball
 * Definition:Bounded Metric Space
 * Definition:Neighborhood (Metric Space)
 * Definition:Open Set/Metric Space
 * Definition:Closed Set/Metric Space
 * Definition:Continuous Mapping (Metric Spaces)
 * Definition:Limit of Sequence/Metric Space
 * Definition:Convergent Sequence (Metric Space)
 * Definition:Cauchy Sequence/Metric Space
 * Definition:Complete Metric Space

Topology

 * Definition:Topology
 * Definition:Topological Space
 * Definition:Open Set/Topology
 * Definition:Closed Set/Topology, open set
 * Definition:Clopen Set, closed, open
 * Definition:Discrete Topology
 * Definition:Indiscrete Topology
 * Definition:Particular Point Topology
 * Definition:Excluded Point Topology
 * Definition:Separation (Topology)
 * Definition:Topological Subspace

countable basis
 * Definition:Interior (Topology)
 * Definition:Basis (Topology)
 * Definition:Basis (Topology)/Analytic Basis
 * Definition:Local Basis, open neighbourhood
 * Definition:First-Countable Space
 * Definition:Second-Countable Space, countable basis


 * Definition:Neighborhood (Topology)/Point
 * Definition:Open Neighborhood, neighbourhood, open set
 * Definition:Limit Point/Topology/Set, open neighbourhood
 * Definition:Closure (Topology), derived set (/closed set)
 * Definition:Boundary (Topology), interior, closure
 * Definition:Everywhere Dense, closure
 * Definition:Separable Space, everywhere dense
 * Definition:Connected (Topology)/Topological Space, separation
 * Definition:Connected Set (Topology), separated sets
 * Definition:Component (Topology)
 * Definition:Dense-in-itself


 * Definition:Filter
 * Definition:Filter on Set


 * Definition:Cover of Set
 * Definition:Subcover/Finite
 * Definition:Countably Compact Space
 * Definition:Open Cover
 * Definition:Compact Topological Space
 * Definition:Compact Subspace
 * Definition:Weakly Locally Compact Space
 * Definition:Lindelöf Space
 * Definition:Irreducible Space


 * Definition:Continuous Mapping (Topology)
 * Definition:Continuous Mapping (Topology)/Everywhere
 * Definition:Homeomorphism/Topological Spaces


 * Definition:Path (Topology)
 * Definition:Path-Connected/Topology/Set


 * Definition:Kolmogorov Space
 * Definition:Fréchet Space (Topology)
 * Definition:Hausdorff Space
 * Definition:T3 Space
 * Definition:T4 Space
 * Definition:T5 Space


 * Definition:Product Space (Topology)


 * Definition:Topological Group


 * Definition:Cartesian Product/Cartesian Space/Real Cartesian Space
 * Definition:Euclidean Space
 * Definition:Euclidean Space/Real
 * Definition:Euclidean Space/Euclidean Topology/Real Number Line
 * Definition:Vector (Euclidean Space)


 * Definition:Topology Induced by Metric
 * Definition:Sequentially Compact Space


 * Definition:Borel Sigma-Algebra

Analysis

 * Definition:Supremum of Set/Real Numbers
 * Definition:Infimum of Set/Real Numbers


 * Definition:Extended Real Number Line


 * Definition:Real Function
 * Definition:Real-Valued Function
 * Definition:Strictly Increasing/Real Function
 * Definition:Strictly Decreasing/Real Function
 * Definition:Bounded Mapping/Real-Valued


 * Definition:Convergent Sequence
 * Definition:Convergent Real Sequence
 * Definition:Limit of Sequence/Real Numbers
 * Definition:Series
 * Definition:Power Series
 * Definition:Generating Function
 * Definition:Convergent Series
 * Definition:Convergent Series/Number Field
 * Definition:Absolutely Convergent Series
 * Definition:Uniform Convergence


 * Definition:Cauchy Sequence


 * Definition:Continuous on Interval
 * Definition:Continuous Real Function
 * Definition:Continuous Complex Function


 * Definition:Exponential Function/Real
 * Definition:Natural Logarithm
 * Definition:Complex Natural Logarithm


 * Definition:Gamma Function


 * Definition:Differentiable Mapping/Real Function/Interval
 * Definition:Derivative
 * Definition:Partial Derivative
 * Definition:Differentiation
 * Definition:Primitive (Calculus)
 * Definition:Primitive (Calculus)/Arbitrary Constant
 * Definition:Integrable Function
 * Definition:Definite Integral
 * Definition:Differential Equation
 * Definition:Differential Equation/Solution
 * Definition:Solution Set of Differential Equation
 * Definition:Initial Condition
 * Definition:Auxiliary Equation
 * Definition:Exact Differential Equation


 * Definition:First Order Ordinary Differential Equation
 * Definition:Second Order Ordinary Differential Equation
 * Definition:Constant Coefficient Homogeneous Linear Second Order ODE


 * Definition:Smooth Path/Complex
 * Definition:Open Set/Complex Analysis
 * Definition:Functional/Real
 * Definition:Riemann Zeta Function


 * Definition:Laplace Transform


 * Definition:Hilbert Space

To be Categorised

 * Definition:Strictly Increasing/Mapping
 * Definition:Odd Function
 * Definition:Extended Real-Valued Function
 * Definition:Divergent Series
 * Definition:Sufficiently Large
 * Definition:Complex Exponential Function


 * Definition:Differentiable Mapping/Real Function
 * Definition:Derivative/Higher Derivatives/Second Derivative
 * Definition:Series/Sequence of Partial Sums
 * Definition:Limit of Real Function
 * Definition:Compact Space


 * Definition:Contour (Complex Plane)
 * Definition:Linear First Order Ordinary Differential Equation
 * Definition:Holomorphic Function
 * Definition:Subdivision (Real Analysis)/Finite
 * Definition:Derivative/Higher Derivatives/Higher Order
 * Definition:Analytic Function

Geometry

 * Definition:Geometry


 * Definition:Geometric Figure
 * Definition:Point
 * Definition:Line
 * Definition:Line/Endpoint
 * Definition:Line/Straight Line
 * Definition:Line/Segment
 * Definition:Line/Curve
 * Definition:Rational Line Segment
 * Definition:Bisection
 * Definition:Intersection (Geometry)
 * Definition:Geometric Mean/Mean Proportional


 * Definition:Linear Measure
 * Definition:Linear Measure/Length
 * Definition:Linear Measure/Breadth
 * Definition:Linear Measure/Distance
 * Definition:Commensurable in Length
 * Definition:Angle
 * Definition:Right Angle
 * Definition:Perpendicular
 * Definition:Right Angle/Perpendicular/Plane
 * Definition:Parallel (Geometry)/Lines


 * Definition:Dimension (Geometry)


 * Definition:Position
 * Definition:Locus


 * Definition:Boundary (Geometry)
 * Definition:Geometric Figure/Plane Figure
 * Definition:Polygon
 * Definition:Polygon/Side
 * Definition:Polygon/Vertex
 * Definition:Polygon/Internal Angle
 * Definition:Polygon/Regular
 * Definition:Triangle (Geometry)
 * Definition:Triangle (Geometry)/Right-Angled
 * Definition:Triangle (Geometry)/Isosceles
 * Definition:Triangle (Geometry)/Equilateral
 * Definition:Similar Triangles
 * Definition:Quadrilateral/Square
 * Definition:Quadrilateral/Rectangle
 * Definition:Quadrilateral/Rectangle/Containment
 * Definition:Quadrilateral/Parallelogram
 * Definition:Circle
 * Definition:Circle/Center
 * Definition:Circle/Radius
 * Definition:Circle/Diameter
 * Definition:Circle/Circumference
 * Definition:Circle/Arc
 * Definition:Unit Circle
 * Definition:Ellipse
 * Definition:Surface
 * Definition:Plane Surface/The Plane
 * Definition:Plane Surface
 * Definition:Area
 * Definition:Rational Area
 * Definition:Medial Area


 * Definition:Sphere/Geometry
 * Definition:Polyhedron
 * Definition:Polyhedron/Face
 * Definition:Polyhedron/Edge
 * Definition:Volume


 * Definition:Sine
 * Definition:Sine/Real Function
 * Definition:Cosine
 * Definition:Cosine/Real Function
 * Definition:Tangent to Curve/Circle
 * Definition:Tangent Function
 * Definition:Cotangent
 * Definition:Secant Function
 * Definition:Cosecant
 * Definition:Hyperbolic Sine
 * Definition:Hyperbolic Cosine
 * Definition:Hyperbolic Tangent
 * Definition:Hyperbolic Cotangent
 * Definition:Hyperbolic Secant

Graph Theory

 * Definition:Graph (Graph Theory)
 * Definition:Graph (Graph Theory)/Vertex
 * Definition:Degree (Vertex)
 * Definition:Graph (Graph Theory)/Edge
 * Definition:Graph (Graph Theory)/Order


 * Definition:Directed Graph
 * Definition:Simple Graph
 * Definition:Connected Graph
 * Definition:Tree (Graph Theory)
 * Definition:Tree (Graph Theory)/Node
 * Definition:Tree (Graph Theory)/Leaf Node


 * Definition:Rooted Tree
 * Definition:Rooted Tree/Root Node
 * Definition:Cycle (Graph Theory)
 * Definition:Path (Graph Theory)

Category Theory

 * Definition:Category
 * Definition:Object (Category Theory)
 * Definition:Morphism
 * Definition:Category of Sets
 * Definition:Identity Morphism
 * Definition:Functor/Covariant
 * Definition:Commutative Diagram


 * Definition:Metacategory

Probability

 * Definition:Sigma-Algebra
 * Let $X$ be a set. A $\sigma$-algebra $\mathcal R$ over $X$ is a system of subsets of $X$ with the following properties:
 * $(SA 1)$ Unit: $X \in \mathcal R$
 * $(SA 2)$ Closure under Complement: $\forall A \in \mathcal R : \complement_X \left({A}\right) \in \mathcal R$
 * $(SA 3)$ Closure under Countable Unions: $\forall A_n \in \mathcal R: n = 1, 2, \ldots : \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R$
 * Definition:Sigma-Algebra Generated by Collection of Subsets
 * Let $X$ be a set. Let $\mathcal G \subseteq \powerset X$ be a collection of subsets of $X$. The $\sigma$-algebra generated by $\mathcal G$, $\map \sigma {\mathcal G}$, is the smallest $\sigma$-algebra on $X$ that contains $\mathcal G$.


 * Definition:Measurable Space
 * Let $\Sigma$ be a $\sigma$-algebra on a set $X$. Then the pair $\left({X, \Sigma}\right)$ is called a measurable space.
 * Definition:Measure (Measure Theory)
 * Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\mu: \Sigma \to \overline{\R}$ be a mapping, where $\overline{\R}$ denotes the set of extended real numbers. Then $\mu$ is called a measure on $\Sigma$ iff it has the following properties:
 * $(1): \quad$ For every $E \in \Sigma$: $\mu \left({E}\right) \ge 0$
 * $(2): \quad$ For every sequence of pairwise disjoint sets $\left\{{S_{n}}\right\} \subseteq \Sigma$: $\displaystyle \mu \left({\bigcup_{n \mathop = 1}^{\infty} S_n}\right) = \sum_{n \mathop = 1}^{\infty} \mu \left({S_{n}}\right)$ (that is, $\mu\ $ is a countably additive function).
 * $(3): \quad$ There exists at least one $E \in \Sigma$ such that $\mu \left({E}\right)$ is finite.
 * Definition:Measurable Set
 * Let $\left({X, \Sigma}\right)$ be a measurable space. A subset $S \subseteq X$ is said to be ($\Sigma$-)measurable $S \in \Sigma$.
 * Definition:Measurable Function
 * Let $\left({X, \Sigma}\right)$ be a measurable space. Let $E \in \Sigma$. Then a function $f: E \to \R$ is said to be $\Sigma$-measurable on $E$ iff: $\forall \alpha \in \R: \left\{{x \in E : f \left({x}\right) \le \alpha}\right\} \in \Sigma$
 * Definition:Measure Space
 * A measure space is a triple $\left({X, \Sigma, \mu}\right)$ where:
 * $X$ is a set
 * $\Sigma$ is a $\sigma$-algebra on $X$
 * $\mu$ is a measure on $\Sigma$.


 * Definition:Probability Space
 * A probability space is a measure space $\left({\Omega, \Sigma, \Pr}\right)$ in which $\Pr \left({\Omega}\right) = 1$. A probability space is used to define the parameters determining the outcome of an experiment $\mathcal E$. In this context, the elements of a probability space are generally referred to as follows:
 * $\Omega$ is called the sample space of $\mathcal E$
 * $\Sigma$ is called the event space of $\mathcal E$
 * $\Pr$ is called the probability measure on $\mathcal E$.
 * Definition:Experiment
 * Definition:Event Space
 * Definition:Event
 * Definition:Probability


 * Definition:Random Variable
 * Definition:Random Variable/Discrete
 * Definition:Random Variable/Continuous


 * Definition:Probability Density Function
 * Definition:Expectation
 * Definition:Game
 * Definition:Game/Player
 * Definition:Move


 * Definition:Probability Generating Function
 * Definition:Moment Generating Function

Physics

 * Definition:Physics
 * Definition:Ordinary Space
 * Definition:Time
 * Definition:Physical System
 * Definition:Body
 * Definition:Mass
 * Definition:Force
 * Definition:Velocity
 * Definition:Magnitude
 * Definition:Unit of Measurement
 * Definition:Imperial
 * Definition:SI Units
 * Definition:Metre
 * Definition:Time/Unit/Year
 * Definition:Time/Unit/Day