hcju/proofwikips huggingface dataset:
Definition:Fréchet Space (Functional Analysis)
Let $\R^\omega$ denote the countable-dimensional real Cartesian space. Let: :$x := \family {x_i}_{i \mathop \in \N} = \tuple {x_0, x_1, x_2, \ldots}$ and: :$y := \family {y_i}_{i \mathop \in \N} = \tuple {y_0, y_1, y_2, \ldots}$ denote arbitrary elements of $\R^\omega$. Let the distance function $d: \R^\omega \times \R^\omega \to \R$ be applied to $\R^\omega$ as: :$\forall x, y \in \R^\omega: \map d {x, y} = \displaystyle \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$ The distance function $d$ is referred to as the '''Fréchet (product) metric'''. The resulting metric space $\struct {\R^\omega, d}$ is then referred to as the '''Fréchet (metric) space'''.
Definition:Hilbert Cube
The '''Hilbert cube''' $\struct {I^\omega, d_2}$ is the subspace of the Hilbert sequence space $I^\omega$ defined as: :$\displaystyle I^\omega = \prod_{k \mathop \in \N} \closedint 0 {\dfrac 1 k}$ under the same metric as that of the Hilbert sequence space: :$\displaystyle \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$
Definition:Convex Set (Order Theory)/Definition 1
A subset $A$ of an ordered set $\left({S, \preceq}\right)$ is '''convex (in $S$)''' : :$\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$
Definition:Convex Set (Order Theory)/Definition 2
A subset $A$ of an ordered set $\struct {S, \preceq}$ is '''convex (in $S$)''' : :$\forall x, y \in A: \forall z \in S: x \prec z \prec y \implies z \in A$
Definition:Interval/Open Interval
Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a, b \in S$. === Open Interval on Ordered Set === {{:Definition:Open Interval}} === Open Real Interval === In the context of the real number line $\R$: {{:Definition:Open Real Interval}}
Definition:Interval/Closed Interval
Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a, b \in S$. === Closed Interval on Ordered Set === {{:Definition:Closed Interval}} === Closed Real Interval === In the context of the real number line $\R$: {{:Definition:Closed Real Interval}}
Definition:Interval/Half-Open Interval
Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a, b \in S$. === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}} === Half-Open Real Interval === In the context of the real number line $\R$: {{:Definition:Half-Open Real Interval}}
Definition:Convex Component
Let $\struct {S, \preccurlyeq}$ be an ordered set. The maximally convex sets into which $S$ is partitioned are the '''convex components''' of $S$.
Definition:Interval/Ordered Set/Endpoint
The elements $a, b \in S$ are known as the '''endpoints''' (or '''end points''') of the interval. $a$ is sometimes called the '''left hand endpoint''' and $b$ the '''right hand end point''' of the interval.
Definition:Ordinal Space
Let $\Gamma$ be a limit ordinal. === Open Ordinal Space === {{:Definition:Ordinal Space/Open}} === Closed Ordinal Space === {{:Definition:Ordinal Space/Closed}}
Definition:Uncountable Ordinal
Let $\alpha$ be an ordinal. Then $\alpha$ is said to be '''uncountable'''  it is an uncountable set.
Definition:Ordinal Space/Closed
The '''closed ordinal space on $\Gamma$''' is the set $\closedint 0 \Gamma$ of all ordinal numbers less than or equal to $\Gamma$, together with the order topology.
Definition:Ordinal Space/Open
The '''open ordinal space on $\Gamma$''' is the set $\hointr 0 \Gamma$ of all ordinal numbers (strictly) less than $\Gamma$, together with the order topology.
Definition:Ordinal Space/Open/Countable
Let $\Omega$ denote the first uncountable ordinal. The '''countable open ordinal space on $\Gamma$''' is a particular case of an open ordinal space $\hointr 0 \Gamma$ where $\Gamma < \Omega$.
Definition:Ordinal Space/Closed/Uncountable
Let $\Omega$ denote the first uncountable ordinal. The '''uncountable closed ordinal space on $\Omega$''' is the particular case of a closed ordinal space $\closedint 0 \Gamma$ where $\Gamma = \Omega$.
Definition:Ordinal Space/Closed/Countable
Let $\Omega$ denote the first uncountable ordinal. The '''countable closed ordinal space on $\Gamma$''' is a particular case of an closed ordinal space $\closedint 0 \Gamma$ where $\Gamma < \Omega$.
Definition:Ordinal Space/Open/Uncountable
Let $\Omega$ denote the first uncountable ordinal. The '''uncountable open ordinal space on $\Omega$''' is the particular case of an open ordinal space $\hointr 0 \Gamma$ where $\Gamma = \Omega$.
Definition:Characteristic Polynomial of Element of Algebra
Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$. Let $b \in B$. The '''characteristic polynomial''' of $b$ is the characteristic polynomial of the regular representation $\lambda_b : B \to B$ over $A$.
Definition:Trace of Element of Algebra over Ring
Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$. Let $b \in B$. The '''trace''' $\operatorname{Tr}_{B/A}(b)$ of $b$ is the trace of the regular representation $\lambda_b : B \to B$ over $A$.
Definition:Norm of Element of Algebra over Ring
Let $A$ be a commutative ring with unity. Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module  over $A$. Let $b \in B$. The '''trace''' $N_{B/A}(b)$ of $b$ is the determinant of the regular representation $\lambda_b : B \to B$ over $A$.
Definition:Characteristic Polynomial of Matrix
Let $R$ be a commutative ring with unity. Let $M$ be a square matrix over $R$ of order $n > 0$. Let $I_n$ be the $n\times n$ identity matrix. Let $R[x]$ be the polynomial ring in one variable over $R$. The '''characteristic polynomial''' of $M$ is the determinant of a matrix over $R[x]$: :$p_M (x) = \operatorname{det}(xI - M)$.
Definition:Torsion Element of Module
Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. Let $m \in M$. Then $m$ is a '''torsion element'''  there exists a regular element $a \in R$ with $a m = 0$.
Definition:Torsion Submodule
Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. The '''torsion submodule''' $T(M)$ of $M$ is the submodule of all torsion elements of $M$.
Definition:Torsion Module
Let $R$ be a commutative ring with unity. Let $M$ be a unitary module over $R$. Then $M$ is a '''torsion module'''  every element of $M$ is of torsion, that is, $M$ equals its torsion submodule $T(M)$.
Definition:Zariski Topology on Maximal Spectrum of Ring
Let $A$ be a commutative ring with unity. Let $\operatorname{MaxSpec}(A)$ be its maximal spectrum. === Definition 1 === The '''Zariski topology''' on $\operatorname{MaxSpec}(A)$ is the topology with as closed sets the maximal zero loci. === Definition 2 === The '''Zariski topology''' on $\operatorname{MaxSpec}(A)$ is the subspace topology induced by the Zariski topology on the spectrum $\operatorname{Spec} A$.
Definition:Torsion Subgroup
Let $G$ be an abelian group. Its '''torsion subgroup''' $\map T G$ is the subgroup of all torsion elements.
Definition:Torsion Group
Let $G$ be a group. Then $G$ is a '''torsion group'''  all its elements are torsion elements.
Definition:Purely Inseparable Field Extension
Let $E/F$ be an algebraic field extension. === Definition 1 === The extension $E/F$ is '''purely inseparable'''  every element $\alpha \in E \setminus F$ is inseparable. === Definition 2 === Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable'''  for each $\alpha \in E$ there exists $n \in \N$ such that $\alpha^{p^n} \in F$. === Definition 3 === Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable'''  each element of $E$ has a minimal polynomial of the form $X^{p^n} - a$.
Definition:Initial Homomorphism from Integers to Ring with Unity
Let $\Z$ be the ring of integers. Let $R$ be a ring with unity. The '''initial homomorphism''' $\Z \to R$ is the unital ring homomorphism that sends $n \in \Z$ to the $n$th power of $1$ in $R$: :$ n \mapsto n \cdot 1$.
Definition:Positive Characteristic
Let $R$ be a ring with unity. Then $R$ has '''positive characteristic'''  its characteristic is not $0$.
Definition:Inseparable Element of Field Extension
Let $E/F$ be a field extension. Let $\alpha \in E$ be algebraic over $F$. Then $\alpha$ is '''inseparable'''  it is not separable.
Definition:Relative Separable Closure
Let $E/F$ be a field extension. The '''(relative) separable closure''' of $E$ in $F$ is the intermediate field consisting of all separable elements of $E$ over $F$.
Definition:Inseparable Field Extension
Let $E/F$ be an algebraic field extension. Then $E/F$ is '''inseparable'''  it is not separable.
Definition:Separable Closure
Let $K$ be a field. A '''separable closure''' of $K$ is a separably closed algebraic field extension of $F$. It may be defined as the relative separable closure of $F$ in its algebraic closure.
Definition:Separably Closed Field
Let $K$ be a field. Then $K$ is '''separably closed'''  === Definition 1 === :The only separable field extension of $K$ is $K$ itself. === Definition 2 === :Every separable irreducible polynomial over $K$ has degree $1$. === Definition 3 === :Every separable polynomial over $K$ of strictly positive degree has a root in $K$.
Definition:Separable Degree
Let $E / F$ be a field extension. Let $S \subseteq E$ be the separable closure of $F$ in $E$. The '''separable degree''' $\index E F_{\operatorname {sep} }$ of $E / F$ is the degree $\index S F$.
Definition:Inseparable Degree
Let $E/F$ be a field extension. Let $S \subseteq E$ be the separable closure of $F$ in $E$. The '''inseparable degree''' $[E : F]_i$ of $E/F$ is the degree $[E : S]$.
Definition:Absolute Galois Group
Let $K$ be a field. === Definition 1 === {{:Definition:Absolute Galois Group/Definition 1}} === Definition 2 === {{:Definition:Absolute Galois Group/Definition 2}}
Definition:Profinite Group
=== Definition 1 === A '''profinite group''' is a topological group that is isomorphic to a small inverse limit of finite discrete groups, with the inverse limit topology. === Definition 2 === A '''profinite group''' is a topological group that is compact, Hausdorff and totally disconnected.
Definition:Complete Order Topology
Let $\struct {S, \preceq}$ be a complete ordered set. Let $\tau$ be the order topology on $\struct {S, \preceq}$. Then $\tau$ is a '''complete order topology'''. Hence $\struct {S, \preceq, \tau}$ is a '''complete order space'''.
Definition:Integral Transform
Let $p$ be a variable whose domain is a subset of the set of real numbers $\R$. Let $\closedint a b$ be a closed real interval for some $a, b \in \R: a \le b$. Let $f: \closedint a b \to \R$ be a real function defined on the domain $\closedint a b$. Let $\map K {p, x}$ be a real-valued function defined for all $p$ in its domain and all $x \in \closedint a b$. Let $\map f x \map K {p, x}$ be integrable {{WRT|Integration}} $x$ for all $p$ in its domain and all $x \in \closedint a b$. Consider the real function $\map F p$ defined as: :$\map F p = \displaystyle \int_a^b \map f x \map K {p, x} \rd x$ Then $\map F p$ is an '''integral transform''' of $\map f x$.
Definition:Integral Transform/Kernel
The function $K \left({p, x}\right)$ is the '''kernel''' of $F \left({p}\right)$.
Definition:Integral Transform/Operator
This can be written in the form: :$F = \map T f$ where $T$ is interpreted as the (unitary) operator meaning: :''Multiply this by $\map K {p, x}$ and integrate {{WRT|Integration}} $x$ between the limits $a$ and $b$.'' Thus $T$ transforms the function $\map f x$ into its image $\map F p$, which is itself another real function.
Definition:Integral Transform/Image Space
The domain of $p$ is known as the '''image space''' of $T$.
Definition:Inverse Integral Operator
Let $T: f \to F$ be an integral operator on a function $f$. Let there be a (unitary) operator $T^{-1}: F \to f$ such that for a given $F \left({p}\right)$ there exists a unique $f \left({x}\right)$ such that $f = T \left({f}\right)$. Then $T^{-1}$ is the '''inverse integral operator''' of $T$.
Definition:Inversion Theorem
Let $F \left({p}\right)$ be an integral transform: :$F \left({p}\right) = \displaystyle \int_a^b f \left({x}\right) K \left({p, x}\right) \, \mathrm d x$ Let $T$ be the integral operator associated with $F \left({p}\right)$: :$F = t \left({f}\right)$ An '''inversion theorem''' is a specification for an inverse integral operator $T^{-1}$ of the form $f = T^{-1} \left({F}\right)$ such that: :$f \left({x}\right) = \displaystyle \int_\alpha^\beta F \left({p}\right) H \left({x, p}\right) \, \mathrm d p$ should such an $H \left({x, p}\right)$ exist. It is not necessarily the case that it does exist..
Definition:Trigonometric Series
A '''trigonometric series''' is a series of the type: :$\map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ where: :the domain of $x$ is the set of real numbers $\R$ :the coefficients $a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$ are real numbers independent of $x$.
Definition:Trigonometric Series/Complex Form
A '''trigonometric series''' can be expressed in a form using complex functions as follows: :$\map S x = \displaystyle \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$ where: :the domain of $x$ is the set of real numbers $\R$ :the coefficients $\ldots, c_{-n}, \ldots, c_{-2}, c_{-1}, c_0, c_1, c_2, \ldots, c_n, \ldots$ are real numbers independent of $x$ :$c_{-n} = \overline {c_n}$ where $\overline {c_n}$ is the complex conjugate of $c_n$.
Definition:Fourier Series/Fourier Coefficient/Range 2 Pi
The constants: : $a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$ are the '''Fourier coefficients''' of $f$.
Definition:Dirichlet Conditions
{{begin-axiom}} {{axiom | n = \text D 1         | t = $\quad f$ is absolutely integrable }} {{axiom | n = \text D 2         | t = $\quad f$ has a finite number of local maxima and local minima }} {{axiom | n = \text D 3         | t = $\quad f$ has a finite number of discontinuities, all of them finite }} {{end-axiom}}
Definition:Absolutely Integrable Function
Let $f$ be a real function. $\map f x$ is '''absolutely integrable''' on $S \subseteq \R$  its absolute value of $f$ is an integrable function on $S$. That is,  the definite integral of the absolute value of $f$ over any interval $\openint \alpha \beta \subseteq S$ is bounded.
Definition:Fourier Series/Fourier Coefficient
The constants: : $a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$ are the '''Fourier coefficients''' of $f$.
Definition:Spence's Function
'''Spence's function''', also known as the '''dilogarithm''', is a special case of the polylogarithm, defined for $z \in \C$ by the integral:  :$\displaystyle \operatorname {Li}_2 \paren z = -\int_0^z \frac {\Ln \paren {1 - t} } t \rd t$ where: :$\displaystyle \int_0^z$ is an integral across the straight line in the complex plane connecting $0$ and $z$.  :$\Ln$ is the principal branch of the complex natural logarithm.
Definition:Half-Range Fourier Sine Series
=== Formulation 1 === {{:Definition:Half-Range Fourier Sine Series/Formulation 1}} === Formulation 2 === {{:Definition:Half-Range Fourier Sine Series/Formulation 2}}
Definition:Half-Range Fourier Cosine Series
=== Formulation 1 === {{:Definition:Half-Range Fourier Cosine Series/Formulation 1}} === Formulation 2 === {{:Definition:Half-Range Fourier Cosine Series/Formulation 2}}
Definition:Fourier Series/Formulation 1
Let $\alpha \in \R$ be a real number. Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 \lambda} \map f x \rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = a_n       | r = \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \cos \frac {n \pi x} \lambda \rd x }} {{eqn | l = b_n       | r = \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \sin \frac {n \pi x} \lambda \rd x }} {{end-eqn}} Then: :$\displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$ is the '''Fourier Series''' for $f$.
Definition:Fourier Series/Formulation 2
Let $a, b \in \R$ be real numbers. Let $f: \R \to \R$ be a function such that $\displaystyle \int_a^b \map f x \rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = A_m       | r = \dfrac 2 {b - a} \int_a^b \map f x \cos \frac {2 m \pi \paren {x - a} } {b - a} \rd x }} {{eqn | l = B_m       | r = \dfrac 2 {b - a} \int_a^b \map f x \sin \frac {2 m \pi \paren {x - a} } {b - a} \rd x }} {{end-eqn}} Then: :$\displaystyle \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$ is the '''Fourier Series''' for $f$.
Definition:Half-Range Fourier Sine Series/Formulation 2
Let $\map f x$ be a real function defined on the interval $\openint a b$. Then the ''' half-range Fourier sine series''' of $\map f x$ over $\openint a b$ is the series: :$\displaystyle \map f x \sim \sum_{m \mathop = 1}^\infty B_m \sin \frac {m \pi \paren {x - a} } {b - a}$ where for all $n \in \Z_{> 0}$: :$B_m = \displaystyle \frac 2 {b - a} \int_a^b \map f x \sin\frac {m \pi \paren {x - a} } {b - a} \rd x$
Definition:Half-Range Fourier Cosine Series/Formulation 2
Let $\map f x$ be a real function defined on the interval $\openint a b$. Then the ''' half-range Fourier cosine series''' of $\map f x$ over $\openint a b$ is the series: :$\displaystyle \map f x \sim \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty A_m \cos \frac {m \pi \paren {x - a} } {b - a}$ where for all $m \in \Z_{\ge 0}$: :$A_m = \displaystyle \frac 2 {b - a} \int_a^b \map f x \cos \frac {m \pi \paren {x - a} } {b - a} \rd x$
Definition:Orthonormal Set of Real Functions
Let $I$ be an indexing set. Let $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ be an indexed family of real functions all of which are integrable over the interval $\openint a b$. Let $S$ have the property that: :$\forall m, n \in I: \displaystyle \int_a^b \map {\phi_m} x \map {\phi_n} x \rd x = \delta_{m n}$ where $\delta_{m n}$ denotes the Kronecker delta. Then $S$ is defined as '''orthonormal'''.
Definition:Complete Orthonormal Set of Real Functions
Let $I$ be an indexing set. Let $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ be an orthonormal set of real functions over the interval $\openint a b$. Let $S$ have the property that: :$\forall n \in I: \displaystyle \int_a^b \map \psi x \map {\phi_n} x \rd x = 0 \implies \map \psi x \equiv 0$ for any real function $\psi$ integrable over the interval $\openint a b$. Then $S$ is a '''complete orthonormal set''' of real functions.
Definition:Divisor of Polynomial
Let $D$ be an integral domain. Let $D \sqbrk x$ be the polynomial ring in one variable over $D$. Let $f, g \in D \sqbrk x$ be polynomials. Then: :'''$f$ divides $g$''' :'''$f$ is a divisor of $g$''' :'''$f$ is a factor of $g$''' :'''$g$ is divisible by $f$''' : :$\exists h \in D \sqbrk x : g = f h$ This is denoted: :$f \divides g$ === Notation === {{:Definition:Divisor Notation}}
Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1
The element $y$ is a '''monomial''' of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$  there exists a mapping $a: I \to \N$ with finite support such that: :$y = \ds \prod_{i \mathop \in I} X_i^{a_i}$ where: :$\prod$ denotes the product with finite support over $I$ :$X_i^{a_i}$ denotes the $a_i$th power of $X_i$.
Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2
The element $y$ is a '''monomial''' of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$  there exists a finite set $S$ and a mapping $f: S \to \set {X_i : i \in I}$ such that it equals :$y = \ds \prod_{s \mathop \in S} \map f s$ where $\prod$ denotes the product over the finite set $S$.
Definition:Inclusion-Preserving Mapping
Let $A$ and $B$ be sets. Let $f : A \to B$ be a mapping. Then $f$ is '''inclusion-preserving'''  for every two sets $a_1, a_2 \in A$: :$a_1 \subseteq a_2 \implies f(a_1) \subseteq f(a_2)$
Definition:Topologically Nilpotent Ring Element
Let $A$ be a Hausdorff topological ring. Let $a \in A$. Then $a$ is '''topologically nilpotent'''  the sequence of powers $a^n$ converges to $0$.
Definition:Induced Mapping on Spectra of Rings
Let $A$ and $B$ be commutative rings with unity. Let $f : A \to B$ be a ring homomorphism. The '''induced mapping on spectra by $f$''' is the mapping $f^* : \operatorname{Spec} B \to \operatorname{Spec} A$ between their spectra with: :$f^* (\mathfrak p) = f^{-1}(\mathfrak p)$, the preimage of a prime ideal $\mathfrak p \in \operatorname{Spec} B$. === Induced morphism of locally ringed spaces ===
Definition:Saturation of Ideal by Multiplicatively Closed Subset
Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq A$ be an ideal. Let $S \subseteq A$ be a multiplicatively closed subset. === Definition 1 === The '''saturation''' of $\mathfrak a$ by $S$ is the ideal: :$\{ a \in A : \exists s \in S : as \in \mathfrak a\}$ === Definition 2 === Let $A \overset \iota \to A_S$ be the localization of $A$ at $S$. The '''saturation''' of $\mathfrak a$ is the preimage of its image under the ring homomorphism $\iota : A \to A_S$: :$\iota^{-1}(\iota(\mathfrak a))$
Definition:Topological Closed Embedding
Let $X$ and $Y$ be topological spaces. Let $f : X \to Y$ be a mapping. Then $f$ is a '''(topological) closed embedding''' : *$f$ is a topological embedding *its image $f(X)$ is closed in $Y$
Definition:Topological Open Embedding
Let $X$ and $Y$ be topological spaces. Let $f : X \to Y$ be a mapping. Then $f$ is a '''(topological) open embedding''' : *$f$ is a topological embedding *its image $f(X)$ is open in $Y$
Definition:Unit Ideal of Ring
Let $R$ be a ring. Its '''unit ideal''' is the ideal $R$.
Definition:Semi-Local Ring
Let $A$ be a commutative ring with unity. Then $A$ is '''semi-local'''  $A$ has a finite number of maximal ideals.
Definition:Local Ring/Commutative
Let $A$ be a commutative ring with unity. === Definition 1 === {{Definition:Local Ring/Commutative/Definition 1}} === Definition 2 === {{Definition:Local Ring/Commutative/Definition 2}}
Definition:Localization of Module
Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset. Let $A_S$ be the localization of $A$ at $S$. Let $M$ be an $A$-module. A '''localization of $M$ at $S$''' is a pair $(M_S, \iota)$ where: *$M_S$ is an $A_S$-module *$\iota : M \to M_S$ is the '''localization map''', an $A$-module homomorphism to the restriction of scalars of $M_S$ to $A$ Such that: :For every $A_S$-module $N$ and $A$-module homomorphism $f: M \to \operatorname{res}_A^{A_S}N$ to the restriction of scalars to $A$, there exists a unique $A_S$-module homomorphism $g : M_S \to N$ such that $f = g \circ \iota$. That is, the precomposition mapping between modules of homomorphisms: :$\operatorname{Hom}_{A_S} (M_S, N) \overset {\iota^*} \to \operatorname{Hom}_A(M, N)$ is a bijection.
Definition:Precomposition Mapping
Let $A, B, C$ be sets. Let $\operatorname{Hom}(B, C)$ denote the set of all mappings from $B$ to $C$. Let $f : A \to B$ be a mapping. The '''precomposition mapping''' $f^* : \operatorname{Hom}(B, C) \to \operatorname{Hom}(A, C)$ is the mapping that sends a mapping $g : B \to C$ to the precomposition $g \circ f$ with $f$.
Definition:Vanishing Ideal of Set of Prime Ideals
Let $A$ be a commutative ring with unity. Let $V \subseteq \operatorname{Spec} A$ be a set of prime ideals of $A$. Its '''vanishing ideal''' is its intersection, the set of elements of $A$ that are in each $\mathfrak p \in V$: :$I(V) = \bigcap V$
Definition:Vanishing Ideal of Subset of Affine Space
Let $k$ be a field. Let $n \ge 0$ be a natural number. Let $k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$. Let $S \subseteq \mathbb A^n_k$ be a subset of the standard affine space over $k$. Its '''vanishing ideal''' is the ideal: :$\map I S = \set {f \in k \sqbrk {X_1, \ldots, X_n} : \forall x \in S : \map f x = 0}$