hcju/proofwikips huggingface dataset:
Definition:Completion (Normed Division Ring)
Let $\struct {R_1, \norm {\, \cdot \,}_1}$ and $\struct {R_2, \norm {\, \cdot \,}_2}$ be normed division rings. Let $M_1 = \struct {R_1, d_1}$ and $M_2 = \struct {R_2, d_2}$ be the metric spaces where $d_1: R_1 \times R_1 \to \R_{\ge 0}$ and $d_2: R_2 \times R_2 \to \R_{\ge 0}$ are the metrics induced by $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively. Then '''$\struct {R_2, \norm {\, \cdot \,}_2}$ is a completion of $\struct {R_1, \norm {\, \cdot \,}_1}$''' : :$(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$ :$(2): \quad M_2$ is a metric completion of $\map \phi {M_1}$. That is, '''$\struct {R_2, \norm{\,\cdot\,}_2}$ is a completion of $\struct {R_1,\norm{\,\cdot\,}_1}$''' : :$(a): \quad M_2$ is a complete metric space :$(b): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$ :$(c): \quad \map \phi {R_1}$ is a dense subspace in $M_2$.
Definition:Existential Quantifier/Unique/Also denoted as
The symbol $\exists_1$ is also found for the same concept, being an instance of the exact existential quantifier $\exists_n$. Some sources, for example {{BookReference|Axiomatic Set Theory|1972|Patrick Suppes}}, use $\operatorname E !$, which is idiosyncratic, considering the use in the same source of $\exists$ for the general existential quantifier.
Definition:Order of Group Element/Infinite/Definition 1
$x$ '''is of infinite order''', or '''has infinite order'''  there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$: :$\order x = \infty$
Definition:Order of Group Element/Infinite/Definition 2
$x$ '''is of infinite order''', or '''has infinite order'''  the powers $x, x^2, x^3, \ldots$ of $x$ are all distinct: :$\order x = \infty$
Definition:Order of Group Element/Finite/Also known as
An '''element of finite order''' of $G$ is also known as a '''torsion element''' of $G$.
Definition:Order of Group Element/Finite/Definition 1
$x$ '''is of finite order''', or '''has finite order'''  there exists $k \in \Z_{> 0}$ such that $x^k = e_G$.
Definition:Order of Group Element/Finite/Definition 2
$x$ '''is of finite order''', or '''has finite order'''  there exist $m, n \in \Z_{> 0}$ such that $m \ne n$ but $x^m = x^n$.
Definition:Complete Normed Division Ring
A normed division ring $\struct {R, \norm {\, \cdot \,} }$ is '''complete'''  the metric space $\struct {R, d}$ is a complete metric space where $d$ is the metric induced by the norm $\norm {\, \cdot \,}$. That is, a normed division ring $\struct {R, \norm {\, \cdot \,} }$ is '''complete'''  every Cauchy sequence is convergent.
Definition:Order of Group Element/Infinite/Definition 3
$x$ '''is of infinite order''', or '''has infinite order'''  the group $\gen x$ generated by $x$ is of infinite order. :$\order x = \infty \iff \order {\gen x} = \infty$
Definition:Cardinality/Also known as
Some authors prefer the term '''order''' instead of '''cardinality''', particularly in the context of finite sets. {{AuthorRef|Georg Cantor}} used the term '''power''' and equated it with the term '''cardinal number''', using the notation $\overline {\overline M}$ for the '''cardinality''' of $M$. Some sources cut through all the complicated language and call it the '''size'''. Some sources use $\map {\#} S$ (or a variant) to denote '''set cardinality'''. This notation has its advantages in certain contexts, and is used on occasion on this website. Others use $\map C S$, but this is easy to confuse with other uses of the same or similar notation. A clear but relatively verbose variant is $\Card \paren S$ or $\operatorname{card} \paren S$. {{BookReference|Introductory Real Analysis|1968|A.N. Kolmogorov|author2 = S.V. Fomin}} use $\map m A$ for the '''power''' of the set $A$. Further notations are $\map n A$ and $\overline A$.
Definition:Cardinality/Infinite
Let $S$ be an infinite set. The '''cardinality''' $\card S$ of $S$ can be indicated as: :$\card S = \infty$ However, it needs to be noted that this just means that the cardinality of $S$ cannot be assigned a number $n \in \N$. It means that $\card S$ is ''at least'' $\aleph_0$ (aleph null).
Definition:Field of Integers Modulo Prime
Let $p \in \Bbb P$ be a prime number. Let $\Z_p$ be the set of integers modulo $p$. Let $+_p$ and $\times_p$ denote addition modulo $p$ and multiplication modulo $p$ respectively. The algebraic structure $\struct {\Z_p, +_p, \times_p}$ is '''the field of integers modulo $p$'''.
Definition:Conjugate (Group Theory)/Element/Also known as
Some sources refer to the '''conjugate''' of $x$ as the '''transform''' of $x$. Some sources refer to '''conjugacy''' as '''conjugation'''.
Definition:Conjugate (Group Theory)/Element/Also defined as
Some sources define the '''conjugate of $x$ by $a$ in $G$''' as: :$x \sim y \iff \exists a \in G: x \circ a = a \circ y$ or: :$x \sim y \iff \exists a \in G: a^{-1} \circ x \circ a = y$
Definition:Conjugate (Group Theory)/Element/Definition 1
The  '''conjugacy relation''' $\sim$ is defined on $G$ as: :$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x = y \circ a$
Definition:Conjugate (Group Theory)/Element/Definition 2
The  '''conjugacy relation''' $\sim$ is defined on $G$ as: :$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$
Definition:Ring (Abstract Algebra)/Historical Note
According to {{AuthorRef|Ian Stewart}}, in his {{BookLink|Galois Theory|Ian Stewart|ed = 3rd|edpage = Third Edition}} of $2004$, the ring axioms were first formulated by {{AuthorRef|Heinrich Martin Weber}} in $1893$.
Definition:Norm/Ring
Let $\struct {R, +, \circ}$ be a ring whose zero is denoted $0_R$. A '''(submultiplicative) norm''' on $R$ is a mapping from $R$ to the non-negative reals: {{:Definition:Submultiplicative Norm on Ring}}
Definition:Norm/Division Ring
Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$. A '''(multiplicative) norm''' on $R$ is a mapping from $R$ to the non-negative reals: {{:Definition:Multiplicative Norm on Ring}}
Definition:Norm/Vector Space
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$. Let $V$ be a vector space over $R$, with zero $0_V$. A '''norm''' on $V$ is a map from $V$ to the nonnegative reals: :$\norm{\,\cdot\,}: V \to \R_{\ge 0}$ satisfying the '''(vector space) norm axioms''': {{Definition:Norm Axioms (Vector Space)}}
Definition:Norm/Algebra
Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. Let $A$ be an algebra over $R$. A '''norm''' on $A$ is a vector space norm $\norm{\,\cdot\,}: A \to \R_{\ge 0}$ on $A$ as a vector space such that: :for all $a, b \in A: \norm {a b} \le \norm a \cdot \norm b$
Definition:Norm/Unital Algebra
Let $A$ be an unital algebra over $R$ with unit $e$. A '''norm''' on $A$ is an algebra norm $\norm{\,\cdot\,}: A \to \R_{\ge 0}$ such that: :$\norm e = 1$
Definition:Norm/Bounded Linear Transformation
Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a bounded linear transformation. Then the '''norm''' of $A$, denoted $\norm A$, is the real number defined by: :$(1): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$ :$(2): \quad \norm A = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne \mathbf 0_H}$ :$(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$ :$(4): \quad \norm A = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$ These definitions are equivalent, as proved in Equivalence of Definitions of Norm of Linear Transformation.
Definition:Norm/Bounded Linear Functional/Definition 1
The '''norm''' of $L$ is the real number defined as the supremum: :$\norm L = \sup \set {\size {L h}: \norm h_H \le 1}$
Definition:Norm/Bounded Linear Functional/Definition 2
The '''norm''' of $L$ is the supremum: :$\norm{L} = \sup \set{\left|{Lh}\right|: \norm{h}_H = 1}$
Definition:Norm/Bounded Linear Functional/Definition 3
The '''norm''' of $L$ is the supremum: :$\norm L = \displaystyle \sup \set {\frac {\size {L h} } {\norm h _H}: h \in H, h \ne \bszero_H}$
Definition:Norm/Bounded Linear Functional/Definition 4
The '''norm''' of $L$ is the infimum: :$\norm{L} = \inf \set{c > 0: \forall h \in H: \left|{Lh}\right| \le c \norm{h}_H}$
Definition:Norm/Bounded Linear Functional
Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$. === Definition 1 === {{:Definition:Norm/Bounded Linear Functional/Definition 1}} === Definition 2 === {{:Definition:Norm/Bounded Linear Functional/Definition 2}} === Definition 3 === {{:Definition:Norm/Bounded Linear Functional/Definition 3}} === Definition 4 === {{:Definition:Norm/Bounded Linear Functional/Definition 4}} {{refactor|These ought to go into their own pages, appropriately linked}} As a consequence of definition $(4)$, we have for all $h \in H$ that $\size {L h} \le \norm L \norm h$. As $L$ is bounded, it is assured that $\norm L < \infty$.
Definition:Group Direct Product/Finite Product
Let $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be groups. Let $\displaystyle G = \prod_{k \mathop = 1}^n G_k$ be their cartesian product. Let $\circ$ be the operation defined on $G$ as: :$\circ := \tuple {g_1, g_2, \ldots, g_n} \circ \tuple {h_1, h_2, \ldots, h_n} = \tuple {g_1 \circ_1 h_1, g_2 \circ_2 h_2, \ldots, g_n \circ_n h_n}$ for all ordered $n$-tuples in $G$. The group $\struct {G, \circ}$ is called the '''(external) direct product''' of $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$.
Definition:Bilateral Symmetry/Axis
Let $F$ be a plane geometric figure. Let $\map R F$ be a reflection in an axis of reflection $AB$ which is a symmetry. Then $AB$ is referred to as an '''axis of symmetry''' of $F$.
Definition:Multiplicative Group of Complex Roots of Unity
Let $n \in \Z$ be an integer such that $n > 0$. Let $U_n := \set {z \in \C: z^n = 1}$ denote the set of complex $n$th roots of unity. Let $\struct {U_n, \times}$ be the algebraic structure formed by $U_n$ under complex multiplication. Then $\struct {U_n, \times}$ is the '''multiplicative group of complex $n$th roots of unity'''.
Definition:Isolated Zero
Let $f: \C \to \C$ be a complex function. Let $\map f {z_0} = 0$ for some $z_0 \in \C$.  We say $z_0$ is an '''isolated zero''' of $f$ if there exists an open ball $B$ containing $z_0$, such that $\map f w \ne 0$ for all $w \in B \setminus \set {z_0}$.
Definition:Conjugacy Action/Subsets
Let $\powerset G$ be the power set of $G$. The '''(left) conjugacy action on subsets''' is the group action $* : G \times \powerset G \to \powerset G$: :$g * S = g \circ S \circ g^{-1}$ The '''right conjugacy action on subsets''' is the group action $* : \powerset G \times G \to \powerset G$: :$S * g = g^{-1} \circ S \circ g$
Definition:Subgroup Action
Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ}$ be a subgroup of $G$. Let $*: H \times G \to G$ be the operation defined as: :$\forall h \in H, g \in G: h * g := h \circ g$ This is the '''subgroup action''' of $H$ on $G$.
Definition:Norm/Ring/Multiplicative
:$\norm {\,\cdot\,}: R \to \R_{\ge 0}$ satisfying the '''(ring) multiplicative norm axioms''': {{:Definition:Multiplicative Norm Axioms}}
Definition:Norm/Ring/Submultiplicative
:$\norm {\,\cdot\,}: R \to \R_{\ge 0}$ satisfying the '''(ring) submultiplicative norm axioms''': {{:Definition:Submultiplicative Norm Axioms}}
Definition:Isometric Isomorphism
Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $d_R$ and $d_S$ be the metric induced by the norms $\norm {\,\cdot\,}_R$ and $\norm {\,\cdot\,}_S$ respectively. Let $\phi:R \to S$ be a bijection such that: :$(1): \quad \phi: \struct {R, d_R} \to \struct {S, d_S}$ is an isometry :$(2): \quad \phi: R \to S$ is a ring isomorphism. Then $\phi$ is called an '''isometric isomorphism'''. The normed division rings $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ are said to be '''isometrically isomorphic'''.
Definition:Cartesian Product/Cartesian Space/Two Dimensions
The '''cartesian $2$nd power of $S$''' is: :$S^2 = S \times S = \set {\tuple {x_1, x_2}: x_1, x_2 \in S}$ The set $S^2$ called a '''cartesian space of $2$ dimensions'''.
Definition:Bijection/Also known as
The terms :'''biunique correspondence''' :'''bijective correspondence''' are sometimes seen for '''bijection'''. Authors who prefer to limit the jargon of mathematics tend to use the term '''one-one and onto mapping''' for '''bijection'''. If a '''bijection''' exists between two sets $S$ and $T$, then $S$ and $T$ are said to be '''in one-to-one correspondence'''. Occasionally you will see the term '''set isomorphism''', but the term '''isomorphism''' is usually reserved for mathematical structures of greater complexity than a set. Some authors, developing the concept of '''inverse mapping''' independently from that of the '''bijection''', call such a mapping '''invertible'''. The symbol $f: S \leftrightarrow T$ is sometimes seen to denote that $f$ is a '''bijection''' from $S$ to $T$. Also seen sometimes is the notation $f: S \cong T$ or $S \stackrel f \cong T$ but this is cumbersome and the symbol has already got several uses. == Technical Note == {{:Definition:Bijection/Technical Note}}
Definition:Normed Division Subring
Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a normed division ring. A '''normed division subring''' of $\struct {R, +, \circ, \norm {\, \cdot \,} }$ is a subset $S$ of $R$ such that $\struct{S, +_S, \circ_S, \norm{\, \cdot \,}_S}$ is a normed division ring where: :$(1) \quad +_S$ is the binary operation $+$ restricted to $S \times S$ :$(2) \quad \circ_S$ is the binary operation $\circ$ restricted to $S \times S$ :$(3) \quad \norm {\, \cdot \,}_S$ is the norm $\norm {\, \cdot \,}$ restricted to $S$.
Definition:Square/Mapping
Let $\struct {S, \circ}$ be an algebraic structure. Let $f: S \to S$ be the mapping from $S$ to $S$ defined as: :$\forall x \in S: \map f x := x \circ x$ This is usually denoted $x^2$: :$x^2 := x \circ x$
Definition:Square/Mapping/Element
A '''square (element of $S$)''' is an element $y$ of $S$ for which: :$\exists x \in S: y = x^2$ Such a $y = x^2$ is referred to as '''the square of $x$'''.
Definition:Square/Function/Definition 1
The '''square (function) on $\F$''' is the mapping $f: \F \to \F$ defined as: :$\forall x \in \F: \map f x = x \times x$ where $\times$ denotes multiplication.
Definition:Square/Function
Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$. === Definition 1 === {{:Definition:Square/Function/Definition 1}} === Definition 2 === {{:Definition:Square/Function/Definition 2}}
Definition:Square/Function/Definition 2
The '''square (function) on $\F$''' is the mapping $f: \F \to \F$ defined as: :$\forall x \in \F: \map f x = x^2$ where $x^2$ denotes the $2$nd power of $x$.
Definition:Square/Function/Integer
The '''(integer) square function''' is the integer function $f: \Z \to \Z$ defined as: :$\forall x \in \Z: \map f x = x^2$
Definition:Ultrametric Space
An '''ultrametric space''' is a metric space $\struct {X, d}$ where $d$ is non-Archimedean.
Definition:Unique up to Isomorphism
Let $\mathbf C$ be a category. Let $S \subseteq \operatorname{Ob}(\mathbf C)$ be a subclass of its objects. The class $S$ is '''unique up to isomorphism'''  for all objects $s,t \in S$ there is a isomorphism from $s$ to $t$.
Definition:Power of Element/Also defined as
Some treatments do not define the '''power of an element''' for a magma whose operation is non-associative.
Definition:Power Associativity
Let $\struct {S, \circ}$ be a magma. $\struct {S, \circ}$ is '''power associative''' : :$\forall a \in S: a \circ \paren {a \circ a} = \paren {a \circ a} \circ a$
Definition:Finitely Generated Group
Let $G$ be a group. $G$ is '''finitely generated'''  $G$ has a generator which is finite.
Definition:Infinite Cyclic Group/Definition 1
An '''infinite cyclic group''' is a cyclic group $G$ such that: :$\forall n \in \Z_{> 0}: n > 0 \implies \nexists a \in G, a \ne e: a^n = e$
Definition:Infinite Cyclic Group/Definition 2
An '''infinite cyclic group''' is a cyclic group $G$ such that: :$\forall a \in G, a \ne e: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$ where $e$ is the identity element of $G$. That is, such that all the powers of $a$ are distinct.
Definition:Subset Product Action/Left
The '''(left) subset product action''' of $G$ is the group action $*: G \times \powerset G \to \powerset G$: :$\forall g \in G, S \in \powerset G: g * S = g \circ S$
Definition:Subset Product Action
Let $\struct {G, \circ}$ be a group. Let $\HH$ be the set of subgroups of $G$. === Left Subset Product Action === {{:Definition:Subset Product Action/Left}} === Right Subset Product Action === {{:Definition:Subset Product Action/Right}}
Definition:Subset Product Action/Right
The '''(right) subset product action''' of $G$ is the group action $*: G \times \powerset G \to \powerset G$: :$\forall g \in G, S \in \powerset G: g * S = S \circ g$
Definition:Generated Subgroup/Definition 2
The '''subgroup generated by $S$''' is the intersection of all subgroups of $G$ containing $S$.
Definition:Group Action on Coset Space
Let $G$ be a group. Let $H$ be a subgroup of $G$. The mapping $*: G \times G / H \to G / H$ by the rule: :$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$ is the '''group action on the (left) coset space''' $G / H$.
Definition:Trivial Quotient Group
Let $G$ be a group. The quotient group: :$G / \set {e_G} \cong G$ where: :$\cong$ denotes group isomorphism :$e_G$ denotes the identity element of $G$. is known as the '''trivial quotient group'''.
Definition:Inner Automorphism Group
Let $G$ be a group. The '''inner automorphism group''' of $G$ is the algebraic structure: :$\struct {\Inn G, \circ}$ where: :$\Inn G$ is the '''set of inner automorphisms''' of $G$ :$\circ$ denotes composition of mappings.
Definition:Zeraoulia Function
The '''Zeraoulia function''' is a new special function see <ref>E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969 </ref> , <ref> E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969. View at Google Scholar · View at MathSciNet</ref>   proposed by Zeraoulia Rafik in 17/02/2017 and have been studied by Zeraoulia Rafik,Alvaro Humberto Salas  Davide L.Ocampo  and published in  International Journal of Mathematics and Mathematical Sciences , <ref> Zeraoulia Rafik, Alvaro H. Salas, and David L. Ocampo, “A New Special Function and Its Application in Probability,” International Journal of Mathematics and Mathematical Sciences, vol. 2018, Article ID 5146794, 12 pages, 2018. https://doi.org/10.1155/2018/5146794</ref>,  it behave like more than error function and it is defined as : :$\displaystyle \map T a = \int_0^a \paren {e^{-x^2} }^{\map \erf x} \rd x$ Mathematica gives for the first $100$ digits: :$\map T \infty = 0 \cdotp 9721069927691785931510778754423911755542721833855699009722910408441888759958220033410678218401258734$ with $\map T 0 = 0$
Definition:Generator of Subgroup/Definition by Predicate
A generator of a subgroup can be defined by a '''predicate'''. For example: :$\gen {x \in G: x^2 = e}$ defines the subgroup of $G$ generated by the elements of $G$ of order $2$.
Definition:Generated Normal Subgroup/Definition 3
The '''normal subgroup generated by $S$''', denoted $\gen {S^G}$,  is the smallest normal subgroup of $G$ containing $S$: :$\gen {S^G} = \gen {x G x^{-1}: x \in G}$
Definition:Nu Function/Sequence
The sequence of values of '''$\nu$ function''': $\map \nu n$ for $n = 1, 2, 3, \ldots$ begins: :$1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, \ldots$
Definition:Extension Problem
Let $H$ and $K$ be groups. The '''extension problem''' is to determine all groups $G$ such that: :$K \trianglelefteq G$ :$G / K \cong H$ where: :$K \trianglelefteq G$ denotes that $K$ is a normal subgroup of $G$ :$G / K$ denotes the quotient group of $G$ by $K$ :$\cong$ denotes group isomorphism.
Definition:Involution (Mapping)/Definition 1
$f: A \to A$ is an '''involution''' precisely when: :$\forall x \in A: \map f {\map f x} = x$ That is: :$f \circ f = I_A$ where $I_A$ denotes the identity mapping on $A$.
Definition:Involution (Mapping)/Definition 3
Let $f: A \to A$ be a mapping on $A$. Then $f$ is an involution  $f$ is both a bijection and a symmetric relation. That is,  $f$ is a bijection such that $f = f^{-1}$.
Definition:Self-Inverse Element/Also known as
The definition of a '''self-inverse element''' is usually made in the context of a group. Some sources refer to such an element as an '''involution'''.
Definition:Characterization Theorem
Let $E$ be a non-abelian finite simple group. Let $u \in E$ be a self-inverse element of $E$. Let $H = \map {C_E} u$ be the centralizer of $u$ in $E$. Let $G$ be a finite simple group with a self-inverse element $t$ such that $H \cong \map {C_G} t$. A '''characterization theorem''' is a theorem that proves there is only one such group type $G$. That is, that $G \cong E$ necessarily.
Definition:Group of Units/Ring
Let $\struct {R, +, \circ}$ be a ring with unity. Then the set $U_R$ of units of $\struct {R, +, \circ}$ is called the '''group of units''' of $\struct {R, +, \circ}$. This can be denoted explicitly as $\struct {U_R, \circ}$.
Definition:Chess
'''Chess''' is a game for two players. It is played using '''pieces''' placed on a '''chessboard''', where the players take turns to move a piece. A piece is '''captured''' if an opposing piece moves onto the same space. The opposing piece is '''attacking'''. The captured piece is then removed from the board, unless it is a king. It is forbidden to capture your own piece. If a piece could be captured in one move by any piece, it is '''threatened'''. A piece cannot move over another piece, unless it is a knight, or it is castling. If a king is captured, the player who captured it wins immediately. A king is put in '''check''' if it is threatened by any piece. This is '''checkmate''' if the player who owns the king cannot prevent the king from being captured using one move. The only other way to win is if a player indicates that he resigns, traditionally by knocking over his own king.
Definition:Odd Integer/Odd-Times Odd/Sequence
The sequence of '''odd-times odd integers''' begins: :$9, 15, 21, 25, 27, \ldots$
Definition:Completely Metrizable Topology
Let $\struct {S, \tau}$ be a topological space. The space $\struct {S, \tau}$ is said to be '''completely metrizable'''  there exists a metric $d$ such that: :$\struct {S, d}$ is a complete metric space and: :$\tau$ is the topological space induced by the metric $d$. == Also see == {{refactor|Trivial, but the following statement needs to be justified in its own page}} All completely metrizable topologies are in particular metrizable topologies.
Definition:Limit of Sequence/Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n} $ be a sequence in $R$. <section notitle="true" name="LimitDefinition"> Let $\sequence {x_n}$ converge to $x \in R$. Then $x$ is a '''limit of $\sequence {x_n}$ as $n$ tends to infinity''' which is usually written: :$\displaystyle x = \lim_{n \mathop \to \infty} x_n$ </section>
Definition:Null Sequence/Analysis
=== Complex Numbers === {{:Definition:Null Sequence/Complex Numbers}} === Real Numbers === {{:Definition:Null Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Null Sequence/Rational Numbers}}
Definition:Null Sequence/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0_R$. Let $\sequence {x_n}$ be a sequence in $R$ which converges to the limit $0_R$: :$\displaystyle \lim_{n \mathop \to \infty} x_n = 0_R$ Then $\sequence {x_n}$ is called a '''null sequence'''.
Definition:Null Sequence
=== Normed Division Ring === {{:Definition:Null Sequence/Normed Division Ring}}
Definition:Fermat Pseudoprime/Base 4
Let $q$ be a composite number such that $4^q \equiv 4 \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $4$'''.