hcju/proofwikips huggingface dataset:
Definition:Biconnected Set
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a connected set in $T$. Then $H$ is '''biconnected'''  it is not the union of two disjoint non-degenerate connected sets.
Definition:Cut Point
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a connected set in $T$ and let $p \in H$. Let $p \in H$ such that $H \setminus \set p$ is disconnected, where $\setminus$ denotes set difference. Then $p$ is a cut point of $H$.
Definition:Dispersion Point
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a connected set in $T$ and let $p \in H$. Let $p \in H$ such that $H \setminus \set p$ is totally disconnected, where $\setminus$ denotes set difference. Then $p$ is a dispersion point of $H$.
Definition:Continuum (Topology)
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be both compact and connected in $T$. Then $H$ is a '''continuum'''.
Definition:Degenerate Continuum
Let $T = \left({S, \tau}\right)$ be a topological space. A '''degenerate continuum''' of $T$ is a continuum in $T$ containing exactly one element. === Non-Degenerate Continuum === {{:Definition:Degenerate Continuum/Non-Degenerate}}
Definition:Subcontinuum
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a continuum in $T$. Let $K \subseteq H$ also be a continuum in $T$. Then $K$ is a '''subcontinuum''' of $H$. === Proper Subcontinuum === {{:Definition:Subcontinuum/Proper Subcontinuum}}
Definition:Indecomposable Continuum
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a continuum in $T$. Then $H$ is indecomposable  it is not the union of two different non-degenerate proper subcontinua.
Definition:Composant
Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a continuum in $T$. Let $C \subseteq H$ be a subset of $H$. === Composant of a Continuum === {{:Definition:Composant/Continuum}} === Composant of a Point === {{:Definition:Composant/Point}}
Definition:Punctiform Space
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset of $T$. Then $H$ is punctiform if it contains no non-degenerate continua.
Definition:Metric Space Axioms
{{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x \in A         | m = \map d {x, x} = 0 }} {{axiom | n = \text M 2         | q = \forall x, y, z \in A         | m = \map d {x, y} + \map d {y, z} \ge \map d {x, z} }} {{axiom | n = \text M 3         | q = \forall x, y \in A         | m = \map d {x, y} = \map d {y, x} }} {{axiom | n = \text M 4         | q = \forall x, y \in A         | m = x \ne y \implies \map d {x, y} > 0 }} {{end-axiom}}
Definition:Quasimetric
A '''quasimetric''' on a set $X$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x \in A         | m = \map d {x, x} = 0 }} {{axiom | n = \text M 2         | q = \forall x, y, z \in A         | m = \map d {x, y} + \map d {y, z} \ge \map d {x, z} }} {{axiom | n = \text M 4         | q = \forall x, y \in A         | m = x \ne y \implies \map d {x, y} > 0 }} {{end-axiom}} Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset. The difference between a '''quasimetric''' and a metric is that a '''quasimetric''' does not insist that the distance function between distinct elements is commutative, that is, that $\map d {x, y} = \map d {y, x}$. === Quasimetric Space === {{:Definition:Quasimetric/Quasimetric Space}}
Definition:Trivial Character
Let $G$ be a finite abelian group. The character $\chi_0: G \to \C_{\ne 0}$ defined as: :$\forall g \in G: \chi_0 \left({g}\right) = 1$ is the '''trivial character on $G$'''.
Definition:Bounded Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $M' = \struct {B, d_B}$ be a subspace of $M$. === Definition 1 === {{:Definition:Bounded Metric Space/Definition 1}} === Definition 2 === {{:Definition:Bounded Metric Space/Definition 2}}
Definition:Indicator of Group Element
Let $G$ be a finite group. Let $a \in G$. Let $H$ be a subgroup of $G$. The '''indicator of $a$ in $H$''' is the least strictly positive integer $n$ such that $a^n \in H$.
Definition:Nested Sequence
Let $S$ be a set. Let $\SS = \powerset S$ be the power set of $S$. Let $\family {S_k}_{k \mathop \in \N}$ indexed family of subsets of $S$ such that either: :$\forall k \in \N: S_k \subseteq S_{k + 1}$ or: :$\forall k \in \N: S_k \supseteq S_{k + 1}$ Then $\family {S_k}_{k \mathop \in \N}$ is a '''nested sequence''' of sets.
Definition:Topologically Complete Space
Let $T = \struct {S, \tau}$ be a topological space. Let $M = \struct {S, d}$ be a complete metric space such that $\struct {S, \tau}$ is the topological space induced by $d$. If there exists such a complete metric space, then $T$ is described as '''topologically complete'''.
Definition:Sigma-Locally Finite Basis
Let $T = \struct {S, \tau}$ be a topological space. A '''$\sigma$-locally finite basis''' for $T$ is a basis which is the countable union of locally finite covers. {{explain|Is this the same thing as "countably locally finite"?}}
Definition:Quasiuniformity
Let $S$ be a set. A '''quasiuniformity''' on $S$ is a set of subsets $\UU$ of the cartesian product $S \times S$ satisfying the following '''quasiuniformity axioms''': {{:Definition:Quasiuniformity Axioms}} That is, a '''quasiuniformity''' on $S$ is a filter on the cartesian product $S \times S$ (from $(\text U 1)$ to $(\text U 3)$) which also fulfils the condition: :$\forall u \in \UU: \exists v \in \UU$ such that whenever $\tuple {x, y} \in v$ and $\tuple {y, z} \in v$, then $\tuple {x, z} \in u$ which can be seen to be an equivalent statement to $(\text U 4)$. $u \circ v$ in this context can be seen to be equivalent to composition of relations. Thus a '''quasiuniformity''' on $S$ is a filter on $S \times S$ which also fulfils the condition that every element is the composition of another element with itself.
Definition:Uniformity
Let $S$ be a set. A '''uniformity''' on $S$ is a set of subsets $\UU$ of the cartesian product $S \times S$ satisfying the '''quasiuniformity axioms''': {{:Definition:Quasiuniformity Axioms}} and also: :$(\text U 5): \forall u \in \UU: u^{-1} \in \UU$ where $u^{-1}$ is defined as: ::$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$ :That is, all elements of $\UU$ are symmetric.
Definition:Entourage
Let $S$ be a set. Let $\UU \subseteq \powerset {S \times S}$ be a quasiuniformity on $S$. An element $u \in \UU$ of such a quasiuniformity is known as an '''entourage''' of $\UU$'''. Note that an entourage of $\UU$ actually ''is'' a relation on $S$.
Definition:Separated Quasiuniformity
Let $\UU$ be a quasiuniformity on a set $S$. Then $\UU$ is '''separated''' : :$\forall u, v \in \UU: u \cap v = \Delta_S$ That is,  the intersection of all its entourages is the diagonal relation.
Definition:Symmetric Filter Basis
Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. From the definition, a quasiuniformity on $S$ is also a filter on the cartesian product $S \times S$. Let $\BB \subset \powerset {S \times S}$ be a filter basis of $\UU$. Then $\BB$ is a '''symmetric filter basis''' of $\UU$  every element of $\BB$ is symmetric.
Definition:Inverse Entourage
Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. Let $u \in \UU$ be an entourage of $\UU$. Then the '''inverse entourage''' $u^{-1}$ is defined as: :$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$
Definition:Symmetric Entourage
Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. Let $u \in \UU$ be an entourage of $\UU$ such that: :$u = u^{-1}$ where $u^{-1}$ is the inverse of $u$. Then $u$ is '''symmetric'''.
Definition:Quasiuniform Space
Let $\UU$ be a quasiuniformity on a set $S$. Then a topology $\tau$ can be created from $\UU$ by: :$\tau := \set {\map u x: u \in \UU, x \in S}$ where: :$\forall x \in S: \map u x := \set {y: \tuple {x, y} \in u}$ The resulting topological space $T = \struct {S, \tau}$ is called a quasiuniform space. It can be denoted $\struct {\struct {S, \UU}, \tau}$, or just $\struct {S, \UU}$ if it is understood that $\tau$ is the topology created from $\UU$.
Definition:Compatible Quasiuniformities
Let $\UU_1$ and $\UU_2$ be quasiuniformities on a set $S$. Let $\struct {\struct {S, \UU_1}, \tau_1}$ and $\struct {\struct {S, \UU_2}, \tau_2}$ be the quasiuniform spaces generated by $\UU_1$ and $\UU_2$. Then $\UU_1$ and $\UU_2$ are '''compatible (with each other)'''  their topologies are equal. That is,  $\tau_1 = \tau_2$.
Definition:Quasiuniformizable Space
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''quasiuniformizable''' if there exists a quasiuniformity $\UU$ on $S$ such that $\struct {\struct {S, \UU}, \tau}$ is a quasiuniform space.
Definition:Uniform Space
Let $\UU$ be a uniformity on a set $S$. The quasiuniform space $\struct {\struct {S, \UU}, \tau}$ generated from $\UU$ is a '''uniform space'''.
Definition:Uniformizable Space
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''uniformizable''' if there exists a uniformity $\UU$ on $S$ such that $\struct {\struct {S, \UU}, \tau}$ is a uniform space.
Definition:Metrizable Uniformity
Let $M = \struct {A, d}$ be a metric space. Let $\UU$ be the uniformity on $X$ defined as: :$\UU := \set {u_\epsilon: \epsilon \in \R_{>0} }$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\paren {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is defined as '''metrizable'''.
Definition:Superfunction
Let $C, D \subseteq \C$ with $z \in C \implies z + 1 \in C$. Let $F: C \to D$ and $H: D \to D$ be holomorphic functions. Let $\map H {\map F z} = \map F {z + 1}$ for all $z \in C$. Then $F$ is said to be a '''superfunction''' of $H$, and $H$ is called a '''transfer function''' of $F$.
Definition:Tetration
500pxrightthumb$y = \operatorname{tet}_b \left({x}\right)$ versus $x$ for various $b$ 500pxrightthumb$f = \operatorname{tet}_b \left({x}\right)$ in the $x, b$ plane with levels $f = \text{const}$. === Definition for Integers === For all $x \in \R$, $n \in \Z_{\ge 0}$: :${}^n x := \begin{cases} 1 & : n = 0 \\ x^{ \left({{}^{n - 1} x}\right)} & : n > 0 \\ \end{cases}$ Using Knuth uparrow notation: :$x \uparrow \uparrow n := \begin{cases} 1 & : n = 0 \\ x \uparrow \left({x \uparrow \uparrow \left({n - 1}\right)}\right) & : n > 0 \\ \end{cases}$ === Definition for base $b \ge \exp \left({1 / e}\right)$ === Let $b \in \R$ such that $b \ge \exp \left({\dfrac 1 e}\right)$. Let $L \in \C$ be a fixed point of $\log_b$ such that $\Im \left({L}\right) \ge 0$. Let $C = \C \setminus \left\{ {x \in \R: x \le -2}\right\}$. Let $\operatorname{tet}_b: C \mapsto \C$ be the superfunction of $z \mapsto b^z$ such that: : $\operatorname{tet}_b \left({0}\right) = 1$ : $\forall z \in C: \operatorname{tet}_b \left({z^*}\right) = \operatorname{tet}_b \left({z}\right)^*$ : $\displaystyle \forall x \in \R: \lim_{y \to +\infty} \operatorname{tet}_b \left({x + \mathrm i y}\right) = L$ Then the function $\operatorname{tet}_b$ is called '''tetration to base $b$'''. === Definition for $0 < b < \exp \left({1 / e}\right)$ === Let $b \in \R$ such that $1 < b < \exp \left({\dfrac 1 e}\right)$. Let $L_1, L_2 \in \R: L_1 < L_2$ be the fixed points of $\log_b$. Let $T = \dfrac{2 \pi i} {\ln \left({L_1 \ln \left({b}\right)}\right)}$ Let $C = \C \setminus \left\{ {x + T m, x \in \R: x \le -2, m \in \Z}\right \}$ Let $\operatorname{tet}_b: C \mapsto \C$ be the superfunction of $z \mapsto b^z$ such that: : $\operatorname{tet}_b (0) = 1$ : $\forall z \in C: \operatorname{tet}_b(z^*) = \operatorname{tet}_b(z)^*$ : $\forall z \in C: \operatorname{tet}_b(z) = \operatorname{tet}_b(z+T)$ : $\displaystyle \forall y \in \R: \lim_{x \to -\infty} \operatorname{tet}_b (x + \mathrm i y) = L_2 ~~ $ : $\displaystyle \forall \varepsilon \in \R_{>0}: \exists X \in \R$ such that: :: $\forall x \in \R: x > X: \left|{\operatorname {tet}_b \left({x + i y}\right) - L_1}\right| < \varepsilon$   Then the function $\operatorname{tet}_b$ is called '''tetration to base $b$'''.
Definition:Abundant Number
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $\map A n$ denote the abundance of $n$. === Definition 1 === {{:Definition:Abundant Number/Definition 1}} === Definition 2 === {{:Definition:Abundant Number/Definition 2}} === Definition 3 === {{:Definition:Abundant Number/Definition 3}}
Definition:Deficient Number
Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Deficient Number/Definition 1}} === Definition 2 === {{:Definition:Deficient Number/Definition 2}} === Definition 3 === {{:Definition:Deficient Number/Definition 3}}
Definition:Discrete Uniformity
Let $S$ be a set. The '''discrete uniformity''' on $S$ is the uniformity $\UU$ defined as: :$\UU := \set {u \subseteq S \times S: \Delta_S \subseteq u}$ that is, all subsets of the cartesian product on $S$ which contain the diagonal relation on $S$.
Definition:Quasiuniformity Axioms
{{begin-axiom}} {{axiom | n = \text U 1         | q = \forall u \in \UU         | m = \Delta_S \subseteq u }} {{axiom | n = \text U 2         | q = \forall u, v \in \UU         | m = u \cap v \in \UU }} {{axiom | n = \text U 3         | q = \forall u \in \UU         | m = u \subseteq v \subseteq S \times S \implies v \in \UU }} {{axiom | n = \text U 4         | q = \forall u \in \UU         | m = \exists v \in \UU: v \circ v \subseteq u }} {{end-axiom}} where: :$\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$ :$\circ$ is defined as: ::$u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$
Definition:Uniformity Axioms
{{begin-axiom}} {{axiom | n = \text U 1         | q = \forall u \in \UU         | m = \Delta_S \subseteq u }} {{axiom | n = \text U 2         | q = \forall u, v \in \UU         | m = u \cap v \in \UU }} {{axiom | n = \text U 3         | q = \forall u \in \UU         | m = u \subseteq v \subseteq S \times S \implies v \in \UU }} {{axiom | n = \text U 4         | q = \forall u \in \UU         | m = \exists v \in \UU: v \circ v \subseteq u }} {{axiom | n = \text U 5         | q = \forall u \in \UU         | m = u^{-1} \in \UU }} {{end-axiom}} where: :$\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$ :$\circ$ is defined as: ::$u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$ :$u^{-1}$ is defined as: ::$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$ :That is, all elements of $\UU$ are symmetric.
Definition:Comparable Topologies
Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$. Then $\tau_1$ and $\tau_2$ are '''comparable'''  either: :$\tau_1$ is coarser than $\tau_2$ or :$\tau_1$ is finer than $\tau_2$ That is, by definition of coarser and finer, either: :$\tau_1 \subseteq \tau_2$ or :$\tau_1 \supseteq \tau_2$
Definition:Partition Topology
Let $S$ be a set. Let $\PP$ be a partition of $S$. Let $\tau$ be the set of subsets of $S$ defined as: :$a \in \tau \iff a$ is the union of sets of $\PP$ Then $\tau$ is a '''partition topology''' on $S$, and $\struct {S, \tau}$ is a '''partition (topological) space'''.
Definition:Trivial Partition
Let $S$ be a set such that $S \ne \varnothing$. There are two partitions on $S$ which are referred to as '''the trivial partitions on $S$''': === Singleton Partition === {{:Definition:Trivial Partition/Singleton}} === Partition of Singletons === {{:Definition:Trivial Partition/Partition of Singletons}}
Definition:Odd-Even Topology
Let $\Z_{>0}$ denote the set of strictly positive integers: :$\Z_{>0} = \set {x \in \Z: x > 0}$ Let $\PP$ be the partition on $\Z_{>0}$ defined as: :$\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$ That is: :$\PP = \set {\set {1, 2}, \set {3, 4}, \set {5, 6}, \ldots}$ Then the topology whose basis is $\PP$ is called the '''odd-even topology'''.
Definition:Deleted Integer Topology
Let $\PP$ be the set: :$\PP = \set {\openint {n - 1} n: n \in \Z_{> 0} }$ that is, the set of all open real intervals of the form: :$\openint 0 1, \openint 1 2, \openint 2 3, \ldots$ Let $S$ be the set defined as: :$S = \displaystyle \bigcup \PP = \R_{\ge 0} \setminus \Z$ that is, the positive real numbers minus the integers. Let $T = \struct {S, \tau}$ be the partition topology whose basis is $\PP$. Then $T$ is called '''the deleted integer topology'''.
Definition:Pseudometrizable Topology
Let $\left({S, d}\right)$ be a pseudometric space. Let $\left({S, \tau_d}\right)$ be the topological space induced by $d$. Then for any topological space which is homeomorphic to such a $\left({S, \tau_d}\right)$, it and its topology are defined as '''pseudometrizable'''.
Definition:Nondeterministic Turing Machine
A '''Nondeterministic''' (or '''Non-deterministic''') '''Turing Machine''' (NTM) is a variation of the classical Turing machine that relaxes the restriction that all the steps in the machine must be definite. That is, given a single internal state and a single character being read on the tape the machine may have more then one possible response. If any sequence of choices puts the machine into a halting state, then the machine stops after $n$ steps, where $n$ is the minimum number of steps needed to put the machine into a halting state.
Definition:Double Pointed Topology
Let $T = \struct {S, \tau_S}$ be a topological space. Let $A = \set {a, b}$ be a doubleton. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on $A$. Let $\struct {T \times D, \tau}$ be the product space of $T$ and $D$. Then $T \times D$ is known as the '''double pointed topology''' on $T$.
Definition:NP Complexity Class
In computation theory, $NP$ is a complexity class in which there exists some nondeterministic Turing machine that will either: : halt within $\map p {\size x}$ steps or: :run forever where $p$ is a polynomial and $\size x$ is the length of the input to the Machine.
Definition:NP-Complete
A problem $L$ is '''NP-complete''' if any problem in the complexity class NP can be reduced in polynomial time and space to $L$.
Definition:Particular 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 $\tau_p$ of the power set $\powerset S$ as: :$\tau_p = \set {A \subseteq S: p \in A} \cup \set \O$ that is, all the subsets of $S$ which include $p$, along with the empty set. Then $\tau_p$ is a topology called the '''particular point topology on $S$ by $p$''', or just '''a particular point topology'''.
Definition:Finitely Generated Field Extension
Let $E / F$ be a field extension. Then $E$ is said to be '''finitely generated over $F$''' , for some $\alpha_1, \ldots, \alpha_n \in E$: :$E = F \left({\alpha_1, \ldots, \alpha_n}\right)$  where $F \left({\alpha_1, \ldots, \alpha_n}\right)$ is the field in $E$ generated by $F \cup \left\{{\alpha_1, \ldots, \alpha_n}\right\}$.
Definition:Primitive Element of Field Extension
Let $F / K$ be a simple field extension such that $F = K \left({\alpha}\right)$. Then $\alpha$ is a '''primitive element''' of $F$.
Definition:Relative Algebraic Closure
Let $L / K$ be a field extension. The '''relative algebraic closure''' of $K$ contained in $L$ is the set of all elements of $L$ that are algebraic over $K$.
Definition:Algebraically Closed Field
Let $K$ be a field. Then $K$ is '''algebraically closed''' : === Definition 1 === {{:Definition:Algebraically Closed/Definition 1}} === Definition 2 === {{:Definition:Algebraically Closed/Definition 2}} === Definition 3 === {{:Definition:Algebraically Closed/Definition 3}}
Definition:Tower of Fields
A '''tower of fields''': :$F_k / F_{k-1} / \dotsb / F_2 / F_1$ is a finite sequence of field extensions: :$F_{j+1} / F_j$ where $j = 1, \dotsc, k-1$.
Definition:Formal Derivative of Polynomial
Let $R$ be a ring. Let $R \left[{X}\right]$ be the polynomial ring over $R$. Let $f = a_0 + a_1 X + \cdots + a_n X^n \in R \left[{X}\right]$. The '''formal derivative''' $f'$ of $f$ is the polynomial: :$f' \left({X}\right) = a_1 + 2 a_2 X + \cdots + n a_n X^{n-1}$
Definition:Symmetric Polynomial
Let $K$ be a field. Let $K \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomial forms over $K$. A polynomial $f \in K \sqbrk {X_1, \ldots, X_n}$ is '''symmetric'''  for every permutation $\pi$ of $\set {1, 2, \ldots, n}$: :$\map f {X_1, \dotsc, X_n} = \map f {X_{\map \pi 1}, \dotsc, X_{\map \pi n} }$
Definition:Elementary Symmetric Polynomial
Let $K$ be a field. Let $K \left[{X_1, \ldots, X_n}\right]$ be the ring of polynomial forms over $K$. The '''elementary symmetric polynomials in $n$ variables''' are: :$\displaystyle f_r \left({X_1, \ldots, X_n}\right) = \sum_{1 \mathop \le i_1 \mathop < \cdots \mathop < i_r \mathop \le n} x_{i_1} \cdots x_{i_r}: \quad r = 1, \ldots, n$
Definition:Splitting Field
=== Of a polynomial === Let $K$ be a field. Let $f$ be a polynomial over $K$. A '''splitting field''' of $f$ over $K$ is a field extension $L / K$ such that: :$f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$ for some $k \in K$, $\alpha_1, \ldots, \alpha_n \in L$. We say that $f$ '''splits''' over $L$. === Of a set of polynomials === Let $K$ be a field. Let $\FF$ be a set of polynomials over $K$. A '''splitting field''' of $\FF$ over $K$ is a field extension $L / K$ such that for any $f \in \FF$: :$f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$ for some $k \in K$, $\alpha_1, \ldots, \alpha_n \in L$. We say that $\FF$ '''splits''' over $L$.
Definition:Sierpiński Space
The '''Sierpiński space''' is a particular point space with exactly two elements. Its usual presentation is: :$T = \struct {\set {0, 1}, \set {\O, \set 0, \set {0, 1} } }$ that is, as a particular point topology on the set $\set {0, 1}$ where the particular point is $0$. It can also immediately be seen to be an excluded point topology on the set $\set {0, 1}$ where the excluded point is $1$. The '''Sierpiński space''' is considered to be a '''trivial instance''' of both the particular point topology and the excluded point topology.
Definition:Closed Extension Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $p$ be a new element for $S$ such that $S^*_p := S \cup \set p$. Let $\tau^*_p$ be the set defined as: :$\tau^*_p := \set {U \cup \set p: U \in \tau} \cup \set \O$ That is, $\tau^*_p$ is the set of all sets formed by adding $p$ to all the open sets of $\tau$ and including the empty set. Then: :$\tau^*_p$ is the closed extension topology of $\tau$ and: :$T^*_p := \struct {S^*_p, \tau^*_p}$ is the closed extension space of $T = \struct {S, \tau}$.