hcju/proofwikips huggingface dataset:
Definition:Disconnected (Topology)/Points
Let $T = \left({S, \tau}\right)$ be a topological space. Let $a, b \in S$. Then $a$ and $b$ are '''disconnected (in $T$)'''  they are not '''connected (in $T$)'''.
Definition:Local Dimension of Topological Space
Let $X$ be a topological space. Let $x \in X$. The '''local dimension of $X$ at $x$''' is the supremum of lengths of chains of closed irreducible sets of $T$ containing $x$, ordered by inclusion. Thus, the '''Krull dimension''' is $\infty$  there exist arbitrarily long chains containing $x$.
Definition:Krull Dimension of Scheme
Let $\struct {X, \mathcal O_X}$ be a scheme. Its '''Krull dimension''' is the Krull dimension of $X$.
Definition:Disconnected (Topology)/Set/Definition 1
$H$ is a '''disconnected set of $T$'''  it is not a '''connected set of $T$'''.
Definition:Disconnected (Topology)/Set/Definition 2
$H$ is a '''disconnected set of $T$'''  there exist open sets $U$ and $V$ of $T$ such that: : $H \subseteq U \cup V$ : $H \cap U \cap V = \O$ : $U \cap H \ne \O$ and: : $V \cap H \ne \O$
Definition:Local Krull Dimension of Scheme
Let $\struct {X, \mathcal O_X}$ be a scheme. Let $x \in X$. The '''local Krull dimension at $x$''' is the local dimension of $X$ at $x$.
Definition:Vanishing Set of Subset of Ring
Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a subset. The '''vanishing set''' of $S$ is the set of prime ideals of $A$ containing $S$: :$V(S) = \{\mathfrak p \in \operatorname{Spec}(A) : \mathfrak p \supseteq S \}$
Definition:Structure Sheaf of Spectrum of Ring
Let $A$ be a commutative ring with unity. Let $\struct {\Spec A, \tau}$ be its spectrum with Zariski topology $\tau$ === Definition 1 === Note that Principal Open Subsets form Basis of Zariski Topology on Prime Spectrum. We define the structure sheaf of $\Spec A$ to be the sheaf induced by a sheaf on this basis defined as follows: :For $f \in A$, $\map \OO {\map X f}$ is the localization of $A$ at $f$ :For $f, g \in A$ with $\map X f \supset \map X g$, the restriction is the induced homomorphism of $A$-algebras $A_f \to A_g$. === Definition 2 === {{definition wanted|the étalé space point of view}} === Definition 3 === {{definition wanted|using explicit formulas}}
Definition:Irreducible Scheme
Let $\struct {X, \mathcal O_X}$ be a scheme. Then $\struct {X, \mathcal O_X}$ is '''irreducible'''  $X$ is an irreducible space.
Definition:Local Ring Homomorphism
Let $(A, \mathfrak m)$ and $(B, \mathfrak n)$ be commutative local rings. Let $f : A \to B$ be a unital ring homomorphism. === Definition 1 === {{Definition:Local Ring Homomorphism/Definition 1}} === Definition 2 === {{Definition:Local Ring Homomorphism/Definition 2}} === Definition 3 === {{Definition:Local Ring Homomorphism/Definition 3}}
Definition:Quasicompact Scheme
Let $\struct {X, \mathcal O_X}$ be a scheme. Then $X$ is '''quasicompact'''  $X$ is a compact space.
Definition:Preadditive Category
A '''preadditive category''' is a monoidally enriched category over the monoidal category of abelian groups $\mathbf {A b}$. That is, a category such that: :its hom sets are abelian groups and where: :composition is bilinear.
Definition:Sequence of Partial Quotients
Let $F$ be a field. Let $n \in \N\cup\{\infty\}$ be an extended natural number. Let $C$ be a continued fraction in $F$ of length $n$. The '''sequence of partial quotients''' of $C$ is just $C$ itself. That is, a continued fraction equals its '''sequence of partial quotients'''.
Definition:Degenerate Connected Set/Topological Space
$T$ is a '''degenerate connected space'''  it contains exactly one element.
Definition:Canonical Form
A '''canonical form''' of a mathematical object is a standard way of presenting that object as a mathematical expression.
Definition:Ordered Set of Natural Numbers
The '''ordered set of natural numbers''' is the ordered set $(\N, \leq)$ where: :$\N$ is the set of natural numbers :$\leq$ is the ordering on the natural numbers
Definition:Extended Natural Numbers
Let $\N$ be the set of natural numbers. === Definition 1 === The '''extended natural numbers''' are the elements of the ordered set $\struct {\N \cup \set \infty, \le}$, where :$\infty$ is a new element, called '''infinity''' :$\le$ is the ordering on the extended natural numbers === Definition 2 === The '''extended natural numbers''' are the order completion of the ordered set of natural numbers.
Definition:Length of Continued Fraction
Let $k$ be a field. Let $C$ be a continued fraction in $k$, either finite or infinite. The '''length''' of $C$ is an extended natural number equal to: *$\infty$ if $C$ is an infinite continued fraction. *$n$ if $C$ is a finite continued fraction with domain the integer interval $\left[0 \,.\,.\, n\right]$.
Definition:Nonzero Ring Element
Let $R$ be a ring. A '''nonzero element''' of $R$ is an element that is not its ring zero.
Definition:Divisor-Finite Monoid
Let $(M, *)$ be a monoid. Then $M$ is '''divisor-finite'''  for all $m \in M$ the set: :$\{(x, y) \in M^2 : x*y = m \}$ is finite.
Definition:Convolution of Mappings on Divisor-Finite Monoid
Let $\struct {M, \cdot}$ be a divisor-finite monoid. Let $\struct {R, +, \times}$ be a non-associative ring. Let $f, g : M \to R$ be mappings. The '''convolution''' of $f$ and $g$ is the mapping $f * g: M \to R$ defined as: :$\forall m \in M: \map {\paren {f * g} } m := \displaystyle \sum_{x y \mathop = m} \map f x \times \map g y$ where the summation is over the finite set $\set {\tuple {x, y} \in M^2: x y = m}$.
Definition:Big Monoid Ring
Let $(M,\cdot)$ be a divisor-finite monoid. Let $(R, +, \times)$ be an additive semiring. The '''big monoid ring''' of $R$ over $M$ is the ringoid $(R^M, +, *)$ where: :$R^M$ be the set of all mappings $M \to R$ :$+$ is the pointwise operation induced by $+$ :$*$ denotes convolution of mappings
Definition:Generic Point of Topological Space
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$ be an element of $S$. === Definition 1 === The point $x$ is a '''generic point of $T$'''  the closure of the singleton $\left\{{x}\right\}$ is $S$. === Definition 2 === The point $x$ is a '''generic point of $T$'''  $x$ is contained in every non-empty open subset of $T$.
Definition:Sober Space
Let $T = \left({S, \tau}\right)$ be a topological space. Then $T$ is a '''sober space'''  each closed irreducible subspace of $T$ has a unique generic point.
Definition:Ring of Formal Power Series
Let $R$ be a commutative ring with unity. === One variable === A '''ring of formal power series''' over $R$ is a pointed algebra over $R$, that is, an ordered triple $\left({RX, \iota, X}\right)$ where: :$RX$ is a commutative ring with unity :$\iota : R \to RX$ is a unital ring homomorphism, called canonical embedding :$X$ is an element of $RX$, called indeterminate that may be defined as follows: Let $\N$ be the additive monoid of natural numbers. Let $R\N$ be the big monoid ring of $R$ over $\N$. Let $\iota : R \to R\N$ be the embedding. Let $X \in R\N$ be the mapping $X : \N \to R$ defined by $X(n) = 1$ if $n=1$ and $X(n) = 0$ otherwise. The '''ring of formal power series''' over $R$ is the ordered triple $\left({R\N, \iota, X}\right)$ === Multiple variables ===
Definition:Ring of Formal Laurent Series
A '''ring of formal Laurent series in one variable''' over $R$ is a pointed algebra over $R$, that is, an ordered triple $\tuple {\map R {\paren X}, \iota, X}$ where: :$\map R {\paren X}$ is a commutative ring with unity :$\iota: R \to \map R {\paren X}$ is a unital ring homomorphism, called canonical embedding :$X$ is an element of $\map R {\paren X}$, called variable that may be defined as follows: {{explain|Research that double parenthesis notation and investigate its validity and specific meaning}} Let $\tuple {R \sqbrk {\sqbrk X}, \kappa, X}$ be a ring of formal power series in one variable over $R$. Let $\tuple {\map R {\paren X}, \lambda}$ be the localization of $R \sqbrk {\sqbrk X}$ at $X$. The '''ring of formal Laurent series''' over $R$ is the ordered triple $\tuple {\map R {\paren X}, \lambda \circ \kappa, \map \lambda X}$.
Definition:Ring of Formal Dirichlet Series
The '''ring of formal Dirichlet series''' is the topological ring $(\mathcal A, \tau)$ where: :$\mathcal A$ is the ring of arithmetic functions :$\tau$ is the topology on formal dirichlet series
Definition:Field of Formal Laurent Series
Let $k$ be a field. The ring of formal Laurent series $k((X))$ over $k$ in a variable $X$ is also called the '''field of formal Laurent series'''.
Definition:Continued Fraction Expansion of Laurent Series
Let $k$ be a field. Let $k((t^{-1}))$ be the field of formal Laurent series in the variable $t^{-1}$. === Irrational Laurent series === Let $f \in k((t^{-1}))$ be an irrational formal Laurent series. The '''continued fraction expansion''' of $f$ is the infinite continued fraction $(\lfloor \alpha_n \rfloor)_{n\geq0}$ where $\alpha_n$ is recursively defined as: :$\alpha_n = \displaystyle \begin{cases} f & : n = 0 \\  \dfrac 1 {f - \lfloor f \rfloor} & : n \geq 1 \end{cases}$ where $\lfloor \cdot \rfloor$ denotes the polynomial part. === Rational Laurent series ===
Definition:Irreducible Space/Definition 6
A topological space $T = \struct {S, \tau}$ is '''irreducible'''  the closure of every non-empty open set is the whole space: :$\forall U \in \tau: U^- = S$
Definition:Ultraconnected Space/Definition 1
A topological space $T = \struct {S, \tau}$ is '''ultraconnected'''  no two non-empty closed sets are disjoint.
Definition:Ultraconnected Space/Definition 2
A topological space $T = \struct {S, \tau}$ is '''ultraconnected'''  the closures of every distinct pair of elements of $S$ are not disjoint: :$\forall x, y \in S: \set x^- \cap \set y^- \ne \O$
Definition:Irreducible Space/Definition 7
A topological space $T = \struct {S, \tau}$ is '''irreducible'''  every open set of $T$ is connected.
Definition:Locally Connected Space/Definition 4
A topological space $T = \struct {S, \tau}$ is '''locally connected'''  the components of the open sets of $T$ are also open in $T$.
Definition:Convergent of Continued Fraction/Definition 1
The '''$k$th convergent''' $C_k$ of $C$ is the value of the finite continued fraction: :$C_k = \left[{a_0, a_1, \ldots, a_k}\right]$
Definition:Convergent of Continued Fraction/Definition 2
The '''$k$th convergent''' $C_k$ of $C$ is the quotient of the $k$th numerator $p_k$ by the $k$th denominator $q_k$: :$C_k = \dfrac{p_k}{q_k}$
Definition:New Element
Let $S$ be a set. A '''new element for $S$''' is a set that is not an element of $S$. By Set does not Contain Itself, it may be defined as $S$ itself.
Definition:Locally Path-Connected Space/Definition 4
A topological space $T = \struct{S, \tau}$ is a '''locally path-connected space'''  the path components of open sets of $T$ are also open in $T$.
Definition:Weakly Locally Connected at Point/Definition 1
The space $T$ is '''weakly locally connected at $x$'''  $x$ has a neighborhood basis consisting of connected sets.
Definition:Weakly Locally Connected at Point/Definition 2
The space $T$ is '''weakly locally connected at $x$'''  every open neighborhood $U$ of $x$ contains an open neighborhood $V$ such that every two points of $V$ lie in some connected subset of $U$.
Definition:Fundamental Group Functor
Let $\mathbf{Top}^*$ be the category of pointed topological spaces. Let $\mathbf{Grp}$ be the category of groups. The '''fundamental group functor''' is the covariant functor $\pi_1 : \mathbf{Top}^* \to \mathbf{Grp}$ with: {{DefineFunctor |ob = A pointed topological space $(X, x_0)$ is sent to its fundamental group. |mor = A continuous pointed mapping $f$ is sent to its induced homomorphism of fundamental groups $\pi_1(f) = f_*$. }}
Definition:Pointed Topological Space
A '''pointed topological space''' is an ordered pair $\left({T, x_0}\right)$ where: :$T$ is a topological space :$x_0$ is a point of $T$.
Definition:Discrete Category on Set
Let $S$ be a set. The '''discrete category on $S$''' is the discrete category $\mathbf{Dis} \left({S}\right)$ with: {{DefineCategory |ob = elements of $S$ |mor = only identity mappings |comp = composition of mappings }}
Definition:Graded Object
Let $C$ be a category. Let $S$ be a set. Let $\mathbf{Dis}(S)$ be its discrete category. An '''$S$-graded object''' of $C$ is a covariant functor $\mathbf{Dis}(S) \to C$.
Definition:Binary Biproduct
Let $A$ be a category. Let $a_1, a_2$ be objects of $A$. === Definition 1 === A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ such that: :$\tuple {a_1 \oplus a_2, p_1, p_2}$ is a binary product :$\tuple {a_1 \oplus a_2, i_1, i_2}$ is a binary coproduct === Definition 2: for categories with zero morphisms === Let $A$ be a category with zero morphisms. Then $a_1$ and $a_2$ are said to '''have a biproduct''' : :they have a coproduct $\tuple {a_1 \sqcup a_2, j_1, j_2}$ and a product $\tuple {a_1 \times a_2, q_1, q_2}$ :the canonical mapping $r: a_1 \sqcup a_2 \to a_1 \times a_2$ is an isomorphism in which case: :$\tuple {a_1 \sqcup a_2, j_1, j_2, q_1 \circ r, q_2 \circ r}$ :$\tuple {a_1 \times a_2, r \circ j_1, r \circ j_2, q_1, q_2}$ are '''biproducts''' of $a_1$ and $a_2$. === Definition 3: for preadditive categories === Let $A$ be a preadditive category. A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ where: :$a \oplus a_2$ is an object of $A$ :$i_1 : a_1 \to a_1 \oplus a_2$ :$i_2 : a_1 \to a_1 \oplus a_2$ :$p_1 : a_1 \oplus a_2 \to a_2$ :$p_2 : a_1 \oplus a_2 \to a_2$ are morphisms such that: :$p_1 \circ i_1 = 1_{a_1}$ :$p_2 \circ i_2 = 1_{a_2}$ :$i_1 \circ p_1 + i_2 \circ p_2 = 1_{a_1 \oplus a_2}$ where $1$ denotes the identity morphism. === Definition 4: for preadditive categories === Let $A$ be a preadditive category. A '''biproduct''' of $a_1$ and $a_2$ is an ordered tuple $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ where: :$a \oplus a_2$ is an object of $A$ :$i_1 : a_1 \to a_1 \oplus a_2$ :$i_2 : a_1 \to a_1 \oplus a_2$ :$p_1 : a_1 \oplus a_2 \to a_2$ :$p_2 : a_1 \oplus a_2 \to a_2$ are morphisms such that: :$p_1 \circ i_1 = 1_{a_1}$ :$p_2 \circ i_2 = 1_{a_2}$ :$p_1 \circ i_2 = 0_{a_1}$ :$p_2 \circ i_1 = 0_{a_2}$ :$i_1 \circ p_1 + i_2 \circ p_2 = 1_{a_1 \oplus a_2}$ where: :$1$ denotes the identity morphism :$0$ denotes the zero morphism.
Definition:Zero Morphism in Preadditive Category
Let $A$ be a preadditive category. Let $a,b$ be objects of $A$. The '''zero morphism''' $0 : a \to b$ is the identity element of their hom set $\operatorname{Hom}(a, b)$.
Definition:Additive Category
Let $A$ be a preadditive category. === Definition 1: binary biproducts === The category $A$ is an '''additive category''' : * it has a zero object * it has all binary biproducts. === Definition 2: binary products === The category $A$ is an '''additive category''' : * it has a zero object * it has all binary products. === Definition 3: binary coproducts === The category $A$ is an '''additive category''' : * it has a zero object * it has all binary coproducts. === Definition 4: finite biproducts === The category $A$ is an '''additive category'''  it has all finite biproducts. === Definition 5: finite products === The category $A$ is an '''additive category'''  it has all finite products. === Definition 6: finite coproducts === The category $A$ is an '''additive category'''  it has all finite coproducts.
Definition:Canonical Mapping from Coproduct to Product
Let $C$ be a category with zero morphisms. Let $a, b$ be objects of $C$. Assume they have a coproduct $(a \sqcup b, i_1, i_2)$ and a product $(a \times b, p_1, p_2)$. === Definition 1 === Let: :$j_1 : a \to a \times b$ be the unique morphism such that: :*$p_1 \circ j_1 = 1 : a \to a$ :*$p_2 \circ j_1 = 0 : a \to b$ :$j_2 : b \to a \times b$ be the unique morphism such that: :*$p_1 \circ j_2 = 0 : b \to a$ :*$p_2 \circ j_2 = 1 : b \to b$ The '''canonical mapping''' from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that: :$r \circ i_1 = j_1$ :$r \circ i_2 = j_2$ === Definition 2 === Let: :$q_1 : a \sqcup b \to a$ be the unique morphism such that: :*$q_1 \circ j_1 = 1 : a \to a$ :*$q_1 \circ j_2 = 0 : b \to a$ :$q_2 : a \sqcup b \to b$ be the unique morphism such that: :*$q_2 \circ j_1 = 0 : a \to a$ :*$q_2 \circ j_2 = 1 : b \to a$ The '''canonical mapping''' from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that: :$p_1 \circ r = q_1$ :$p_2 \circ r = q_2$ {{definition wanted|for arbitrarily many objects}}
Definition:Totally Separated Space/Definition 1
A topological space $T = \struct {S, \tau}$ is '''totally separated''' : :For every $x, y \in S: x \ne y$ there exists a separation $U \mid V$ of $T$ such that $x \in U, y \in V$.
Definition:Totally Separated Space/Definition 2
A topological space $T = \struct {S, \tau}$ is '''totally separated'''  each of its quasicomponents is a singleton set.
Definition:Pre-Abelian Category
=== Definition 1 === A '''pre-abelian category''' is an additive category in which every morphism has a kernel and a cokernel. === Definition 2 === A '''pre-abelian category''' is an additive category with all finite limits and finite colimits.
Definition:Abelian Category
=== Definition 1 === An '''abelian category''' is a pre-abelian category in which: * every monomorphism is a kernel * every epimorphism is a cokernel === Definition 2 === An '''abelian category''' is a pre-abelian category in which: * every monomorphism is the kernel of its cokernel * every epimorphism is the cokernel of its kernel === Definition 3 === An '''abelian category''' is a pre-abelian category in which for every morphism $f$, the canonical morphism from its coimage to its image $\operatorname{coim}(f) \to \operatorname{im}(f)$ is an isomorphism.
Definition:Extremally Disconnected Space/Definition 1
A $T_2$ (Hausdorff) topological space $T = \struct {S, \tau}$ is '''extremally disconnected'''  the closure of every open set of $T$ is open.
Definition:Extremally Disconnected Space/Definition 3
A $T_2$ (Hausdorff) topological space $T = \struct {S, \tau}$ is '''extremally disconnected'''  the closures of every pair of open sets which are disjoint are also disjoint.
Definition:Extremally Disconnected Space/Definition 2
A $T_2$ (Hausdorff) topological space $T = \struct {S, \tau}$ is '''extremally disconnected'''  the interior of every closed set of $T$ is closed.
Definition:Isometry (Metric Spaces)/Definition 1
Let $M_1 = \tuple {A_1, d_1}$ and $M_2 = \tuple {A_2, d_2}$ be metric spaces or pseudometric spaces. Let $\phi: A_1 \to A_2$ be a bijection such that: :$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$ Then $\phi$ is called an '''isometry'''. That is, an '''isometry''' is a distance-preserving bijection.
Definition:Isometry (Metric Spaces)/Definition 2
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces or pseudometric spaces. :$M_1$ and $M_2$ are '''isometric'''  there exist inverse mappings $\phi: A_1 \to A_2$ and $\phi^{-1}: A_2 \to A_1$ such that: ::$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$ :and: ::$\forall u, v \in A_2: \map {d_2} {u, v} = \map {d_1} {\map {\phi^{-1} } u, \map {\phi^{-1} } v}$
Definition:Isometry (Metric Spaces)/Into
Let $\phi: A_1 \to A_2$ be an injection such that: : $\forall a, b \in A_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$ Then $\phi$ is called an '''isometry (from $M_1$) ''into'' $M_2$'''.
Definition:Presheaf on Topological Space/Definition 1
A $\mathbf C$-valued '''presheaf''' on $T$ is a pair $\left({\mathcal F, \operatorname{res} }\right)$ where: : $\mathcal F$ is a mapping on $\tau$ whose image consists of objects of $\mathbf C$ : $\operatorname{res}$ is a mapping on $\left\{ {\left({U, V}\right) \in \tau^2: U \supseteq V}\right\}$ such that for all $U, V, W \in \tau$ with $U \supseteq V \supseteq W$: :: $\operatorname{res}_V^U$ is a morphism from $\mathcal F \left({U}\right)$ to $\mathcal F \left({V}\right)$ :: $\operatorname{res}_U^U = \operatorname{id}_{\mathcal F \left({U}\right)}$, the identity morphism on $\mathcal F \left({U}\right)$ :: $\operatorname{res}_V^U \circ \operatorname{res}_W^V = \operatorname{res}_W^U$, where $\circ$ is the composition in $\mathbf C$
Definition:Presheaf on Topological Space/Definition 2
Let $\tau$ be the category of open sets of $T$. A $\mathbf C$-valued '''presheaf''' on $T$ is a contravariant functor $\tau \to \mathbf C$.
Definition:Section of Étalé Space/Definition 1
A '''section''' of $\map {\operatorname {\acute Et} } {\mathcal F}$ on $U$ is a continuous mapping $s: U \to \map {\operatorname {\acute Et} } {\mathcal F}$ such that: :$\pi \circ s = I_U$ where $I_U$ is the identity mapping on $U$.
Definition:Section of Étalé Space/Definition 2
A '''section''' of $\map {\operatorname {\acute Et} } {\mathcal F}$ on $U$ is a mapping $s: U \to \map {\operatorname {\acute Et} } {\mathcal F}$ such that: :for all $x \in U$ there exists an open neighborhhood $V$ of $x$ in $U$ such that the restriction of $s$ to $V$ is the section associated to some $t \in \map {\mathcal F} V$.
Definition:Direct Image of Presheaf/Definition 1
The '''direct image''' of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_* \mathcal F$ on $T_2$ with: : $f_* \mathcal F \left({V}\right) = \mathcal F \left({f^{-1} \left({V}\right)}\right)$ for all open $V \subset T_2$ : Restrictions $\operatorname{res}^U_V = \operatorname{res}^{f^{-1} \left({U}\right)}_{f^{-1} \left({V}\right)}$
Definition:Direct Image of Presheaf/Definition 2
The '''direct image''' of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_*\mathcal F$ that is the composition $\mathcal F \circ \operatorname{Open} \left({f}\right)$, where $\operatorname{Open}$ is the open subsets functor.
Definition:Sheaf on Topological Space/Definition 1
A $\mathbf C$-valued '''sheaf''' $\mathcal F$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subseteq S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$: :$\left({\mathcal F \left({U}\right), \left({\operatorname{res}^U_{U_i} }\right)_{i \mathop \in I} }\right)$ is the limit of the restriction of $\mathcal F$ to $\left\{ {U}\right\} \cup \left\{ {U_i: i \in I}\right\} \cup \left\{ {U_i \cap U_j : \left({i, j}\right) \in I^2}\right\}$
Definition:Sheaf on Topological Space/Definition 2
Let $\mathbf C$ be a complete category. {{definition wanted|using equalizers}}
Definition:Sheaf on Topological Space/Definition 3
Let $\mathbf C$ be a complete abelian category. A $\mathbf C$-valued '''sheaf''' $\mathcal F$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subset S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$ the sequence: :$0 \to \mathcal F \left({U}\right) \to \displaystyle \prod_{i \mathop \in I} \mathcal F \left({U_i}\right) \to \prod_{\left({i ,j}\right) \mathop \in I^2} \mathcal F \left({U_i \cap U_j}\right)$ is exact. {{explain|what the maps are}} {{definition wanted|using Definition:étalé Space of Presheaf, another via Definition:Sheaf of Sets}}
Definition:Edge Cut
Let $G$ be a graph. An '''edge cut''' of $G$ is a set of edges $W \subseteq \map E G$ such that the edge deletion $G \setminus W$ is disconnected.
Definition:Particular Point
Let $T = \struct {S, \tau}$ be a topological space. A '''particular point of $T$''' is an element $p \in S$ which is distinguished in some manner from the other elements of $S$.
Definition:Euclidean Space/Euclidean Topology/Real Number Line
Let $\R$ denote the real number line. Let $d: \R \times \R \to \R$ denote the Euclidean metric on $\R$. Let $\tau_d$ denote the topology on $\R$ induced by $d$. The topology $\tau_d$ induced by $d$ is called the '''Euclidean topology'''. Hence $\struct {\R, \tau_d}$ is referred to as the '''real number line with the Euclidean topology'''.
Definition:Euclidean Space/Euclidean Topology/Real Number Plane
Let $\R^n$ be an $n$-dimensional real vector space. Let $M = \left({\R^2, d}\right)$ be a real Euclidean space of $2$ dimensions. The topology $\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''. The space $\left({\R^2, \tau_d}\right)$ is known as the '''(real) Euclidean plane'''.