hcju/proofwikips huggingface dataset:
Definition:Ultraproduct
Let $\mathcal L$ be a first-order language and let $I$ be an infinite set. Let $\mathcal U$ be an ultrafilter on $I$. Let $\mathcal M_i$ be an $\mathcal L$-structure for each $i\in I$ The '''ultraproduct''': :$\mathcal M := \displaystyle \left({\prod_{i \mathop \in I} \mathcal M_i }\right) / \mathcal U$ is an $\mathcal L$-structure defined as follows: :$(1): \quad$ The universe of $\mathcal M$: Let $X$ be the cartesian product: :$\prod_{i \mathop \in I} \mathcal M_i$ Define an equivalence relation $\sim$ on $X$ by: :$\left({a_i}\right)_{i \mathop \in I} \sim \left({b_i}\right)_{i \mathop \in I}$  $\left\{ {i \in I: a_i = b_i}\right \} \in \mathcal U$ The universe of $\mathcal M$ is the set of equivalence classes of $X$ modulo $\sim$. These are essentially sequences taken modulo the equivalence relation above, and are sometimes denoted $\left({m_i}\right)_\mathcal U$. :$(2): \quad$ Interpretation of non-logical symbols of $\mathcal L$ in $\mathcal M$: For each constant symbol $c$, we define $c^\mathcal M$ to be $\left({c^{\mathcal M_i} }\right)_\mathcal U$. For each $n$-ary function symbol $f$, we define $f^\mathcal M$ by setting: :$f^\mathcal M \left({\left({m_{1, i} }\right)_\mathcal U, \dotsc, \left({m_{n, i} }\right)_\mathcal U}\right)$ to be: :$\left({f^{\mathcal M_i} \left({m_{1, i}, \dotsc, m_{n, i} }\right)}\right)_\mathcal U$ For each $n$-ary relation symbol $R$, we define $R^\mathcal M$ to be the set of $n$-tuples: :$\left({\left({m_{1, i} }\right)_\mathcal U, \dots, \left({m_{n, i} }\right)_\mathcal U}\right)$ from $\mathcal M$ such that: :$\left\{ {i \in I: \left({m_{1, i}, \dotsc, m_{n, i} }\right) \in R^\mathcal M_i}\right\} \in \mathcal U$
Definition:Pure Transcendental Extension
Let $K$ be a field. A '''pure transcendental extension''' is a field extension $L / K$ such that: :for some set $\left\{ {X_\alpha: \alpha \in A}\right\}$, $L$ is the field of rational functions in the indeterminates $\left\{ {X_\alpha}\right\}$.
Definition:Algebraically Independent
Let $L / K$ be a field extension. Let $A \subseteq L$ be a subset of $L$. Let $\map K {\set {X_\alpha}_{\alpha \mathop \in A} }$ be the Field of Rational Functions in the indeterminates $\set {X_\alpha: \alpha \mathop \in A}$. Then $A$ is '''algebraically independent''' over $K$ if there exists a homomorphism: :$\phi: \map K {set {X_\alpha}_{\alpha \mathop \in A} } \to L$ such that: :$\map \phi {X_\alpha} = \alpha$
Definition:Transcendence Degree
Let $K$ be a field, and let $L/K$ be a field extension of $K$. The '''transcendence degree''' of $L/K$ is the largest cardinality of an algebraically independent subset $A \subseteq L$.
Definition:Successor Ordinal
A '''successor ordinal''' is the successor set of an ordinal.
Definition:Limit Ordinal
An ordinal $\lambda$ is a '''limit ordinal'''  it is neither the zero ordinal nor a successor ordinal.
Definition:Excluded Point Topology
Let $S$ be a set which is non-empty. Let $p \in S$ be some '''particular point''' of $S$. We define a subset $\tau_{\bar p}$ of the power set $\powerset S$ as: :$\tau_{\bar p} = \set {A \subseteq S: p \notin A} \cup \set S$ That is, all the subsets of $S$ which do not include $p$, along with the set $S$. Then $\tau_{\bar p}$ is a topology called the '''excluded point topology on $S$ by $p$''', or just '''an excluded point topology'''.
Definition:Open Extension Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $p$ be a new element which is not in $S$. Let $S^*_p = S \cup \set p$. Let $\tau^*_p$ be the set defined as: :$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$ That is, $\tau^*_{\bar p}$ is the set of all sets formed by taking all the open sets of $\tau$ and adding to them the set $S^*_p$. Then: :$\tau^*_{\bar p}$ is the open extension topology of $\tau$ and: :$T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ is the open extension space of $T = \struct {S, \tau}$.
Definition:Boubaker Polynomials
The Boubaker polynomials are the components of the following sequence of polynomials: {{begin-eqn}} {{eqn | l = B_0 \left({x}\right)       | r = 1      }} {{eqn | l = B_1 \left({x}\right)       | r = x      }} {{eqn | l = B_2 \left({x}\right)       | r = x^2 + 2      }} {{eqn | l = B_3 \left({x}\right)       | r = x^3 + x      }} {{eqn | l = B_4 \left({x}\right)       | r = x^4 - 2      }} {{eqn | l = B_5 \left({x}\right)       | r = x^5 - x^3 - 3x      }} {{eqn | l = B_6 \left({x}\right)       | r = x^6 - 2x^4 - 3x^2 + 2      }} {{eqn | l = B_7 \left({x}\right)       | r = x^7 - 3x^5 - 2x^3 + 5x      }} {{eqn | l = B_8 \left({x}\right)       | r = x^8 - 4x^6 + 8x^2 - 2      }} {{eqn | l = B_9 \left({x}\right)       | r = x^9 - 5x^7 + 3x^5 + 10x^3 - 7x      }} {{eqn | o = \vdots}} {{end-eqn}}
Definition:Supremum Norm
Let $S$ be a set. Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space. Let $\mathcal B$ be the set of bounded mappings $S \to X$. For $f \in \mathcal B$ the '''supremum norm''' of $f$ on $S$ is defined as: :$\norm f_\infty = \sup \set {\norm {\map f x}: x \in S}$ === Supremum Norm on Continuous on Closed Interval Real-Valued Functions === {{:Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function}}
Definition:Normed Vector Space
Let $\struct {K, +, \circ}$ be a normed division ring. Let $V$ be a vector space over $K$. Let $\norm {\,\cdot\,}$ be a norm on $V$. Then $\struct {V, \norm {\,\cdot\,} }$ is a '''normed vector space'''.
Definition:Banach Space
A '''Banach space''' is a normed vector space where every Cauchy sequence is convergent.
Definition:Contraction Mapping
Let $\struct {X, d_1}$ and $\struct {Y, d_2}$ be metric spaces. Let $f: X \to Y$ be a mapping. Then $f$ is a '''contraction (mapping)'''  there exists $\kappa \in \R: 0 \le \kappa < 1$ such that: :$\forall x, y \in X: \map {d_2} {\map f x, \map f y} \le \kappa \, \map {d_1} {x, y}$ That is, $f$ is Lipschitz continuous for a Lipschitz constant less than $1$.
Definition:Type
Let $\MM$ be an $\LL$-structure. Let $A$ be a subset of the universe of $\MM$. Let $\LL_A$ be the language consisting of $\LL$ along with constant symbols for each element of $A$. Viewing $\MM$ as an $\LL_A$-structure by interpreting each new constant as the element for which it is named, let $\map {\operatorname {Th}_A} \MM$ be the set of $\LL_A$-sentences satisfied by $\MM$. An '''$n$-type over $A$''' is a set $p$ of $\LL_A$-formulas in $n$ free variables such that $p \cup \map {\operatorname {Th}_A} \MM$ is satisfiable by some $\LL_A$-structure. {{Disambiguate|Definition:Logical Formula}} === Complete Type === We say that an $n$-type $p$ is '''complete (over $A$)''' : :for every $\LL_A$-formula $\phi$ in $n$ free variables, either $\phi \in p$ or $\phi \notin p$. The set of '''complete $n$-types over $A$''' is often denoted by $\map {S_n^\MM} A$. Given an $n$-tuple $\bar b$ of elements from $\MM$, the '''type of $\bar b$ over $A$''' is the complete $n$-type consisting of those $\LL_A$-formulas $\map \phi {x_1, \dotsc, x_n}$ such that $\MM \models \map \phi {\bar b}$. It is often denoted by $\map {\operatorname {tp}^\MM} {\bar b / A}$. === Realization === Given an $\LL_A$-structure $\NN$, a type $p$ is '''realized''' by an element $\bar b$ of $\NN$ : :$\forall \phi \in p: \NN \models \map \phi {\bar b}$. Such an element $\bar b$ of $\NN$ is a '''realization''' of $p$. === Omission === We say that $\NN$ '''omits''' $p$ if $p$ is not realized in $\NN$. Then $p$ is an '''omission''' from $\NN$.
Definition:Linear Recurrence Relation
A '''linear recurrence relation''' is a recurrence relation which has the form: :$a_n y_{n+k} + a_{n-1} y_{n+k-1} + \cdots + a_0 y_k = b \left({k}\right)$ where $a_0, \ldots, a_n$ are constants. === Homogeneous Linear Recurrence Relation === If in a linear recurrence relation $b \left({k}\right) = 0$, then the equation is '''homogeneous'''. Otherwise it is '''inhomogeneous'''.   3nfpbstuhbph1pt4r5euye1w8jnhmok
Definition:Elementary Equivalence
Let $\mathcal{M},\mathcal{N}$ be $\mathcal{L}$-structures. We say that $\mathcal{M}$ and $\mathcal{N}$ are '''elementarily equivalent''' if for all $\mathcal{L}$-sentences $\phi$, we have $\mathcal{M}\models \phi$ if and only if $\mathcal{N}\models \phi$.
Definition:Embedded
Let $V$ and $W$ be normed vector spaces. Then $V$ is '''embedded''' in $W$ : : $V \subseteq W$ The inclusion mapping $i_V : V \to W$ is an '''embedding''' of $V$ into $W$.
Definition:Continuously Embedded
Let $\operatorname{id} : V \to W$ be an embedding of normed vector spaces. Then $V$ is '''continuously embedded''' in $W$ and $\operatorname{id}$ is a '''continuous embedding''' if the identity $\operatorname{id}$ is a continuous mapping. {{explain|Link to the specific context in which continuous mapping is defined}}
Definition:Excluded Set Topology
Let $S$ be a set which is non-empty. Let $H \subseteq S$ be some subset of $S$. We define a subset $\tau_{\bar H}$ of the power set $\powerset S$ as: :$\tau_{\bar H} = \set {A \subseteq S: A \cap H = \O} \cup \set S$ that is, all the subsets of $S$ which are disjoint from $H$, along with the set $S$. Then $\tau_{\bar H}$ is a topology called the '''excluded set topology on $S$ by $H$''', or just '''an excluded set topology'''.
Definition:Either-Or Topology
Let $S = \closedint {-1} 1$ be the closed interval on the real number line from $-1$ to $1$. Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$ such that, for any $H \subseteq S$: :$H \in \tau \iff \paren {\set 0 \nsubseteq H \lor \openint {-1} 1 \subseteq H}$ where $\lor$ is the inclusive-or logical connective. Then $\tau$ is the '''either-or topology''', and $T = \struct {S, \tau}$ is the '''either-or space'''
Definition:Type Space
Let $\MM$ be an $\LL$-structure, and let $A$ be a subset of the universe of $\MM$. Let $\map {S_n^\MM} A$ be the set of complete $n$-types over $A$. The '''space of $n$-types over $A$''' is the topological space formed by the set $\map {S_n^\MM} A$ together with the topology arising from the basis which consists of the sets: :$\sqbrk \phi := \set {p \in \map {S_n^\MM} A:\phi \in p}$ for each $\LL_A$-formula $\phi$ with $n$ free variables. Note that each $\sqbrk \phi$ is also closed in this topology, since $\sqbrk \phi$ is the complement of $\sqbrk {\neg \phi}$ in $\map {S_n^\MM} A$. {{LinkWanted|Definition of the notation $\sqbrk \phi$}}
Definition:Isolated Type
Let $T$ be an $\mathcal L$-theory. Let $\phi \left({\bar v}\right)$ be an $\mathcal L$-formula in $n$ free variables $\bar v$ such that $T \cup \phi \left({\bar v}\right)$ is satisfiable. {{Disambiguate|Definition:Logical Formula}} Let $p$ be an $n$-type of $T$. We say that $\phi$ '''isolates''' $p$ if for all $\psi \in p$, we have: :$T \models \forall \bar{v} \left({ \phi \left({\bar v}\right) \rightarrow \psi \left({\bar v}\right) }\right)$ that is, all $\psi$ are semantic consequences of $\phi$.
Definition:Witness Property
 An $\LL$-theory $T$ is said to have the '''witness property''' if for every $\LL$-formula $\map \phi v$ with one free variable, there is a constant symbol $c$ in $\LL$ such that: {{Disambiguate|Definition:Logical Formula}} :$T \models \paren {\exists v \map \phi v} \to \map \phi c$ that is, $\paren {\exists v \map \phi v} \to \map \phi c$ is a semantic consequence of $T$. That is, every existential statement satisfied by $T$ is witnessed by a constant.
Definition:Theory (Logic)
Let $\mathcal L$ be a logical language. An '''$\mathcal L$-theory''' $T$ is a set of $\mathcal L$-sentences. === Theory of Structure === {{:Definition:Theory (Logic)/Structure}} === Complete Theory === {{:Definition:Theory (Logic)/Complete}} === Maximal Theory === {{:Definition:Theory (Logic)/Maximal}}
Definition:Distance to Nearest Integer Function
The '''distance to nearest integer function''' $\norm \cdot: \R \to \closedint 0 {\dfrac 1 2}$ is defined in the following ways: === Definition 1 === {{:Definition:Distance to Nearest Integer Function/Definition 1}} === Definition 2 === {{:Definition:Distance to Nearest Integer Function/Definition 2}}
Definition:Stability (Model Theory)
Let $T$ be a complete $\mathcal L$-theory where $\mathcal L$ is countable. Let $\kappa$ be an infinite cardinal. === $\kappa$-Stable === {{:Definition:Stability (Model Theory)/Kappa-Stable Theory}} === Stable === {{:Definition:Stability (Model Theory)/Stable Theory}} === Unstable === {{:Definition:Stability (Model Theory)/Unstable Theory}}
Definition:Finite Complement Topology
Let $S$ be a set whose cardinality is usually specified as being infinite. Let $\tau$ be the set of subsets of $S$ defined as: :$H \in \tau \iff \relcomp S H \text { is finite, or } H = \O$ where $\relcomp S H$ denotes the complement of $H$ relative to $S$. Then $\tau$ is the '''finite complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is a '''finite complement space'''.
Definition:Ramanujan Sum
Let $e: \R \to \C$ be the mapping defined as: :$\forall \alpha \in \R: \map e \alpha := \map \exp {2 \pi i \alpha}$ For $q \in \N_{>0}$, $n \in \N$, the '''Ramanujan sum''' is defined as: :$\displaystyle \map {c_q} n = \sum_{\substack {1 \mathop \le a \mathop \le q \\ \gcd \set {a, q} \mathop = 1} } \map e {\frac {a n} q}$
Definition:Countable Complement Topology
Let $S$ be an infinite set whose cardinality is usually taken to be uncountable. Let $\tau$ be the set of subsets of $S$ defined as: :$H \in \tau \iff \relcomp S H$ is countable, or $H = \O$ where $\relcomp S H$ denotes the complement of $H$ relative to $S$. In this definition, countable is used in its meaning that includes finite. Then $\tau$ is the '''countable complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is a '''countable complement space'''.
Definition:Higher-Aleph Complement Topology
Let $S$ be a set whose cardinality is $\aleph_n$ where $n > 0$. Let $\tau \subseteq \powerset S$ be the set of subsets of $S$ defined as: :$\tau = \set {U \subseteq S: \size {\relcomp S U} = \aleph_m: m < n} \cup \set {U \subseteq S: \relcomp S U \text { is finite} } \cup \O$ That is, $\tau$ is the set of subsets of $S$ whose complements relative to $S$ are of a cardinality strictly less than $S$. Then $\tau$ is an '''$\aleph_m$ complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is an '''$\aleph_m$ complement space'''. This construction is an extension of the concept of the finite complement topology and the countable complement topology.
Definition:Saturated Model
Let $T$ be an $\mathcal{L}$-theory. Let $\kappa$ be an infinite cardinal. A model $\mathcal{M}$ of $T$ is $\kappa$-'''saturated''' if for every subset $A$ of the  universe of $\mathcal{M}$ of cardinality strictly less than $\kappa$, and for every  $n\in\mathbb N$, every complete $n$-type $p$ over $A$ is realized in $\mathcal{M}$. That is, $\mathcal{M}$ is $\kappa$-saturated if for all $A  \subseteq \mathcal{M}$ with $A<\kappa$, and for all $n\in\mathbb N$, each $p \in S_{n}^{\mathcal{M}} (A)$ is realized in $\mathcal{M}$. We say $\mathcal{M}$ is '''saturated''' if it is $\kappa$-saturated where $\kappa$ is the cardinality of the universe of $\mathcal{M}$.  1lsodr3oexq9bc7rhx6nzqgdth5h3hq
Definition:Universal Model
Let $T$ be an $\mathcal L$-theory. Let $\kappa$ be an infinite cardinal. A model $\mathcal M$ of $T$ is $\kappa$-'''universal''' : :for every model $\mathcal N$ of $T$ whose universe has cardinality strictly less than $\kappa$, there is an elementary embedding of $\mathcal N$ into $\mathcal M$. That is, $\mathcal M$ is '''$\kappa$-universal''' : :for all models $\mathcal N \models T$ with  cardinality $\left\vert{\mathcal N}\right\vert < \kappa$, there is an elementary embedding $j: \mathcal N \to \mathcal M$. We say $\mathcal M$ is '''universal''' if it is '''$\kappa^+$-universal''' where $\kappa$ is the cardinality of the universe of $\mathcal M$ and $\kappa^+$ is the successor cardinal of $\kappa$. Definitions/Model Theory bklciljojrl7we8ux5qi6os3dtoh5jr
Definition:Categorical (Model Theory)
Let $T$ be an $\mathcal{L}$-theory. Let $\kappa$ be a cardinal. $T$ is $\kappa$-'''categorical''' if whenever $\mathcal{M}$ and $\mathcal{N}$ are models of $T$ of cardinality $\kappa$, then $\mathcal{M}$ and $\mathcal{N}$ are isomorphic.
Definition:Compact Complement Topology
Let $T = \struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Let $\tau^*$ be the set defined as: :$\tau^* = \leftset {S \subseteq \R: S = \O \text { or } \relcomp \R S}$ is compact in $\rightset {\struct {\R, \tau} }$ where $\relcomp \R S$ denotes the complement of $S$ in $\R$. Then $\tau^*$ is the '''compact complement topology''' on $\R$, and $T^* = \struct {\R, \tau^*}$ is the '''compact complement space''' on $\R$.
Definition:Graph (Category Theory)
A '''graph''' is an interpretation of a metagraph within set theory. Let $\mathfrak U$ be a class of sets. A metagraph $\mathcal G$ is a '''graph''' if: : $(1): \quad$ The objects form a subset $\operatorname{vert}\mathcal G \subseteq \mathfrak U$ : $(2): \quad$ The morphisms form a subset $\operatorname{edge}\mathcal G \subseteq \mathfrak U$ If the class $\mathfrak U$ is a set, then morphisms are functions, and the domain and codomain in the definition of a morphism are those familiar from set theory. If $\mathfrak U$ is a proper class this is not the case, for example the morphisms of $\mathcal C$ need not be functions.
Definition:Indiscernible
Let $\mathcal M$ be an $\mathcal L$-structure. Let $I$ be an infinite set. Let $X = \left\{ {x_i \in \mathcal M: i \in I}\right\}$ be an infinite subset of the universe of $\mathcal M$ indexed by $I$. For: : every $n \in \N$ and: : every pair of subsets $\left\{ {i_1, \ldots, i_n}\right\}$ and $\left\{ {j_1, \ldots, j_n}\right\}$ of $I$ each with $n$ distinct elements, let: : $\mathcal M \models \phi \left({x_{i_1}, \ldots, x_{i_n} }\right) \leftrightarrow \phi \left({x_{j_1}, \ldots, x_{j_n} }\right)$ for all $\mathcal L$-formulas $\phi$ with $n$ free variables. Then $X$ is '''(an) indiscernible (set) in $\mathcal M$'''. {{Disambiguate|Definition:Logical Formula}} Informally, $X$ is '''indiscernible (set)''' if $\mathcal M$ cannot distinguish between same-sized ordered tuples over $X$ using $\mathcal L$-formulas.
Definition:Order Indiscernible
Let $\mathcal M$ be an $\mathcal L$-structure. Let $\left({I, \le}\right)$ be an infinite ordered set. Let $X = \left\{ {x_i \in \mathcal M: i \in I}\right\}$ be an infinite subset of the universe of $\mathcal M$ indexed by $I$. Let $A$ be a subset of the universe of $\mathcal M$. $X$ is '''(an) order indiscernible (set) over $A$ in $\mathcal M$''' : : For every $n \in \N$ and every pair of chains $i_1 < \cdots < i_n$ and $j_1 < \cdots < j_n$ in $I$ each with $n$ distinct elements, we have: :: $\mathcal M \models \phi \left({x_{i_1}, \ldots, x_{i_n} }\right) \iff \phi \left({x_{j_1}, \ldots, x_{j_n} }\right)$ : for all $\mathcal L$-formulas $\phi$ with $n$ free variables and parameters from $A$. {{Disambiguate|Definition:Logical Formula}} Informally, $X$ is '''order indiscernible'''  $\mathcal M$ cannot distinguish between same-sized increasing ordered tuples over $X$ using $\mathcal L$-formulas.
Definition:Fort Space
Let $S$ be an infinite set. Let $p \in S$ be a particular point of $S$. Let $\tau_p \subseteq \powerset S$ be a subset of the power set of $S$ defined as: :$\tau_p = \leftset {U \subseteq S: p \in \relcomp S U} \text { or } \set {U \subseteq S: \relcomp S U}$ is finite$\rightset{}$ That is, $\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is finite. Then $\tau_p$ is a '''Fort topology''' on $S$, and the topological space $T = \struct {S, \tau_p}$ is a '''Fort space'''.
Definition:Gödel-Bernays Axioms
The '''Gödel-Bernays axioms''' are a conservative extension of the Zermelo-Fraenkel axioms with the axiom of choice (ZFC) that allow comprehension of classes. Although not the standard axioms of set theory, particularly in category theory they spare us any set of all sets-type paradoxes. == Axioms for Sets == The first five axioms are identical to the axioms of the same names from ZFC. The quantified variables range over the universe of sets. === The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity/Set Theory}} == Axioms for Classes == In the remaining axioms, the quantified variables range over classes. The first two differ from the ZFC axioms with the same names in this way only. The last two have no analogue among the ZFC axioms. === The Axiom of Extension === {{:Axiom:Axiom of Extension (Classes)}} === The Axiom of Foundation === {{:Axiom:Axiom of Foundation (Classes)}} === Class Comprehension === {{:Axiom:Class Comprehension Schema}} === The Axiom of Limitation of Size === {{:Axiom:Axiom of Limitation of Size}}
Definition:Class (Class Theory)
A '''class''' is a collection of all sets such that a particular condition holds. In class builder notation, this is written as: :$\set {x: \map p x}$ where $\map p x$ is a statement containing $x$ as a free variable.   This is read: :'''All $x$ such that $\map p x$ holds.'''
Definition:Metagraph
A '''metagraph''' $\mathcal G$ consists of: * objects $X, Y, Z, \ldots$ * morphisms $f, g, h, \ldots$ between its objects These are subjected to the following two axioms: {{begin-axiom}} {{axiom | n = 1         | lc= ''Domains''         | t = Every morphism $f$ has associated an object $\operatorname{dom} f$, called the '''domain''' of $f$ }} {{axiom | n = 2         | lc= ''Codomains''         | t = Every morphism $f$ has associated an object $\operatorname{cod} f$, called the '''codomain''' of $f$ }} {{end-axiom}} A '''metagraph''' is purely axiomatic, and does not use set theory. For example, the objects are not "elements of the set of objects", because these axioms are (without further interpretation) unfounded in set theory.
Definition:Path (Category Theory)
Let $\mathcal G$ be a graph. A (non-empty) '''path''' in $\mathcal G$ is a sequence $(i,a_1,\ldots,a_n,j)$ such that: *$n > 1$ *$a_1,\ldots, a_n$ are edges of $\mathcal G$ *$i$ is the source of $a_1$ and $j$ is the destination of $a_n$ *For $1 \leq k < n$ the destination of $a_k$ is the source of $a_{k+1}$ This is usually written in the form: :$\displaystyle i \stackrel{a_1}{\longrightarrow} * \stackrel{a_2}{\longrightarrow} \cdots \stackrel{a_{n-1}}{\longrightarrow} * \stackrel{a_n}{\longrightarrow} j$ For any vertex $i$ the '''empty path''' from $i$ to $i$ is the pair $(i,i)$. Paths are '''composed''' by concatenation: :$\displaystyle (j,b_1,\ldots,b_n,k)(i,a_1,\ldots,a_m,j) = (i,a_1,\ldots,a_m,b_1,\ldots,b_n,k)$ Here we have used the somewhat awkward but more common right-to-left notation for composition.  mtbopkxz4y7tctz5weuq28ngbss9blw
Definition:Minimal Topology
Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$. Let $\Theta_S$ be the set of all topologies on $S$: :$\Theta_S = \left\{{\tau \in \mathcal P \left({S}\right): \tau}\right.$ is a topology on $\left.{S}\right\}$ Let $\Phi: \Theta_S \to \left\{{T, F}\right\}$ be a propositional function on $\Theta_S$. Let $\vartheta \in \Theta_S$ have the property that $\Phi \left({\vartheta}\right)$ and: :$\forall \tau \in \Theta_S: \Phi \left({\tau}\right) \implies \vartheta \subseteq \tau$ That is, $\vartheta$ is the coarsest topology on $S$ which satisfies the propositional function $\Phi$. Then $\vartheta$ is the '''minimal topology satisfying $\Phi$'''.  htipkwppnixkuhckesep3patmjqucql