# Category:Explanation Required

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This category is for pages which have a point which needs explaining.

See the Explain template.

## Pages in category "Explanation Required"

The following 200 pages are in this category, out of 836 total.

(previous page) (next page)### A

- Abel's Theorem
- Abel's Theorem/Historical Note
- Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit
- Achilles Paradox
- Definition:Addition of Polynomials
- Definition:Addition of Polynomials/Polynomial Forms
- Definition:Adjoint Linear Transformation
- Axiom:Aleph Hypothesis
- Aleph is Infinite Cardinal
- Definition:Aleph Mapping
- Alexander's Compactness Theorem
- Definition:Algebra (Abstract Algebra)
- Algebraic Element of Field Extension is Root of Unique Monic Polynomial of Minimal Degree
- Alternating Sum and Difference of Binomial Coefficients for Given n
- Alternating Sum and Difference of Binomial Coefficients for Given n/Proof 1
- Alternating Sum and Difference of Factorials to Infinity
- Analytic Continuation of Dirichlet L-Function
- Angle Between Non-Zero Vectors Always Defined
- Angular Velocity is Vector Quantity
- Annihilating Polynomial of Minimal Degree is Irreducible
- Approximation to Reciprocal times Derivative of Gamma Function
- Arc Length of Sector
- Definition:Archimedean Property
- Definition:Archimedean Property/Norm
- Arens-Fort Space is not Locally Connected
- Definition:Argument of Complex Number
- Artin's Theorem on Alternative Algebras
- Atlas Belongs to Unique Differentiable Structure
- Ax-Grothendieck Theorem
- Axiom:Axiom of Choice for Finite Sets
- Axiom:Axiom of Choice for Finite Sets/Proof from Hall's Marriage Theorem
- Axiom of Choice Implies Zorn's Lemma
- Axiom of Choice Implies Zorn's Lemma/Proof 2
- Axiom:Axiom of Continuity
- Axiom:Axiom of Limitation of Size

### B

- Baire-Osgood Theorem
- Banach-Steinhaus Theorem
- Barbier's Theorem
- Definition:Barrier
- Basis for Open Ordinal Topology
- Basis of Free Module is No Greater than Generator
- Bendixson-Dulac Theorem
- Beppo Levi's Theorem
- Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')
- Bertrand's Theorem
- Bertrand-Chebyshev Theorem
- Definition:Big-Omega
- Definition:Bilinear Form
- Definition:Bilinear Functional
- Bolzano-Weierstrass Theorem
- Bolzano-Weierstrass Theorem/Proof 2
- Boolean Prime Ideal Theorem
- Borel-Carathéodory Lemma
- Bound for Analytic Function and Derivatives
- Bounded Function is of Exponential Order Zero
- Bourbaki-Witt Fixed Point Theorem
- Branch of Finite Tree is Finite
- Bézout's Lemma
- Bézout's Lemma/Proof 4

### C

- Canonical Injection into Cartesian Product of Modules
- Canonical Order Well-Orders Ordered Pairs of Ordinals
- Definition:Cantor Normal Form
- Cantor Set is Uncountable
- Cantor Set is Uncountable/Proof 1
- Cantor Space is Nowhere Dense
- Cantor Space is Nowhere Dense/Proof 1
- Cantor-Bernstein-Schröder Theorem
- Cantor-Bernstein-Schröder Theorem/Proof 1
- Cantor-Dedekind Hypothesis
- Carathéodory's Theorem (Analysis)
- Carathéodory's Theorem (Convex Analysis)
- Definition:Card Game with Bluffing
- Cardinal Number is Ordinal
- Cardinal Number Less than Ordinal
- Cardinality of Infinite Union of Infinite Sets
- Carroll Paradox
- Cartesian Product is Small
- Definition:Cartesian Product/Family of Sets/Axiom of Choice
- Definition:Cauchy Functional Equation
- Cauchy Mean Value Theorem
- Cauchy Mean Value Theorem/Geometrical Interpretation
- Cauchy Sequences form Ring with Unity
- Cauchy Sequences form Ring with Unity/Corollary
- Cauchy's Convergence Criterion for Series
- Cauchy's Integral Formula
- Cauchy-Hadamard Theorem/Complex Case
- Cauchy-Hadamard Theorem/Complex Case/Proof 2
- Cauchy-Riemann Equations
- Cauchy-Riemann Equations/Sufficient Condition
- Cayley-Hamilton Theorem/Finitely Generated Modules
- Cayley-Hamilton Theorem/Matrices
- Definition:Center (Abstract Algebra)
- Definition:Center (Abstract Algebra)/Ring
- Definition:Central Field
- Central Field is Field of Functional
- Central Limit Theorem
- Cesàro Mean
- Change of Coordinate Vectors Under Linear Transformation
- Definition:Character (Number Theory)
- Character of Representations over C are Algebraic Integers
- Characterisation of Linearly Independent Set through Free Module Indexed by Set
- Characterisation of Local Rings
- Characterisation of Ordered Fields
- Characteristic of Increasing Mapping from Toset to Order Complete Toset
- Characterization of Measures
- Chebyshev's Inequality/Proof 1
- Chinese Remainder Theorem (Commutative Algebra)
- Class Equality is Reflexive
- Class Equality is Transitive
- Classical Probability is Probability Measure
- Classification of Compact One-Manifolds
- Classification of Compact Two-Manifolds
- Closed Linear Subspaces Closed under Setwise Addition
- Closed Real Interval is Neighborhood Except at Endpoints
- Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets
- Closure of Subset of Double Pointed Topological Space
- CNF Satisfiability Problem is NP-Complete
- Definition:Cofinal Relation on Ordinals
- Collatz Conjecture
- Collatz Conjecture/Supposed Proof
- Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
- Commutative Linear Transformation is G-Module Homomorphism
- Commutativity of Incidence Matrix with its Transpose for Symmetric Design
- Compact Hausdorff Topology is Maximally Compact
- Compact Subspace of Linearly Ordered Space
- Compact Subspace of Linearly Ordered Space/Reverse Implication
- Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 2
- Compactness Properties Preserved under Continuous Surjection
- Compatibility of Atlases is Equivalence Relation
- Complete and Totally Bounded Metric Space is Sequentially Compact
- Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 1
- Definition:Complete Ritz Sequence
- Completion Theorem (Metric Space)/Lemma 4
- Complex Numbers as External Direct Product
- Condition for Agreement of Family of Mappings
- Condition for Cofinal Nonlimit Ordinals
- Condition for Composition Series
- Condition for Differentiable Functional to have Extremum
- Condition for Injective Mapping on Ordinals
- Condition for Set Union Equivalent to Associated Cardinal Number
- Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative
- Conditions for Extremal Embedding in Field of Functional
- Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 2
- Conditions for Strong Minimum of Functional
- Definition:Conjugate Point
- Definition:Conjugate Point/Definition 1
- Definition:Conjugate Point/Definition 3
- Definition:Conjugate Point/wrt original Functional
- Connected Subspace of Linearly Ordered Space
- Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements
- Construction of Smith Number from Prime Repunit
- Definition:Content of Polynomial/GCD Domain
- Definition:Continued Fraction
- Definition:Continued Fraction Expansion
- Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself
- Definition:Continued Fraction/Expansion of Real Number
- Continuous Functions on Compact Space form Banach Space
- Continuous iff Way Below iff There Exists Element that Way Below and Way Below
- Continuous Image of Compact Space is Compact/Corollary 3
- Continuous Image of Compact Space is Compact/Corollary 3/Proof 2
- Continuous Image of Connected Space is Connected
- Continuous Image of Connected Space is Connected/Corollary 3
- Continuous Image of Connected Space is Connected/Proof 1
- Continuous Implies Locally Bounded
- Definition:Continuous Ordered Set
- Continuous Real Function is Darboux Integrable
- Definition:Continuous Total Preordering
- Definition:Continuously Embedded
- Continuum Property
- Continuum Property/Proof 1
- Contour Integral of Gamma Function
- Definition:Convergent Sequence/Rational Numbers
- Cook-Levin Theorem
- Definition:Coset Decoding Table
- Countable Compactness is Preserved under Continuous Surjection
- Countable Sets Have Measure Zero
- Countable Stability implies Stability for All Infinite Cardinalities
- Countable Union of Finite Sets is Countable
- Countably Compact Metric Space is Compact
- Countably Compact Metric Space is Compact/Proof 2
- Cover of Interval By Closed Intervals is not Pairwise Disjoint
- Cowen-Engeler Lemma
- Definition:Critical Point (Topology)
- Cyclic Groups of Order p q
- Cyclic Groups of Order p q/Proof 1
- Cyclicity Condition for Units of Ring of Integers Modulo m
- Cyclotomic Polynomial has Integer Coefficients

### D

- De Morgan's Laws imply Uniquely Complemented Lattice is Boolean Lattice
- Definition:Decomposable Set
- Definition:Dedekind Domain
- Definition:Dedekind Domain/Definition 1
- Dedekind's Theorem
- Dedekind's Theorem/Proof 1
- Definition:Dedekind-MacNeille Completion
- Definite Integral of Uniformly Convergent Series of Continuous Functions
- Definite Integral to Infinity of Reciprocal of 1 plus Power of x
- Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Proof 1
- Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Proof 2
- Definition of Polynomial from Polynomial Ring over Sequence
- Dependent Choice for Finite Sets