hcju/proofwikips huggingface dataset:
Definition:P-adic Valuation/Rational Numbers
Let the $p$-adic valuation on the integers $\nu_p^\Z$ be extended to $\nu_p^\Q: \Q \to \Z \cup \left\{{+\infty}\right\}$ by: :$\nu_p^\Q \left({\dfrac a b}\right) := \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)$ This mapping $\nu_p^\Q$ is called the '''$p$-adic valuation (on $\Q$)''' and is usually denoted $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$.
Definition:Valuation Axioms
{{begin-axiom}} {{axiom | n = \text V 1         | q = \forall a, b \in R         | ml= \map \nu {a \times b}         | mo= =         | mr= \map \nu a + \map \nu b }} {{axiom | n = \text V 2         | q = \forall a \in R         | ml= \map \nu a = +\infty         | mo= \iff         | mr= a = 0_R         | rc= where $0_R$ is the ring zero }} {{axiom | n = \text V 3         | q = \forall a, b \in R         | ml= \map \nu {a + b}         | mo= \ge         | mr= \min \set {\map \nu a, \map \nu b} }} {{end-axiom}}
Definition:Multiplicative Norm Axioms
{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in R         | ml= \norm x = 0         | mo= \iff         | mr= x = 0_R }} {{axiom | n = \text N 2         | lc= Multiplicativity:         | q = \forall x, y \in R         | ml= \norm {x \circ y}         | mo= =         | mr= \norm x \times \norm y }} {{axiom | n = \text N 3         | lc= Triangle Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \norm x + \norm y }} {{end-axiom}}
Definition:Norm Axioms (Vector Space)
{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive definiteness:         | q = \forall x \in V         | ml= \norm x = 0         | mo= \iff         | mr= x = \mathbf 0_V }} {{axiom | n = \text N 2         | lc= Positive homogeneity:         | q = \forall x \in V, \lambda \in R         | ml= \norm {\lambda x}         | mo= =         | mr= \norm {\lambda}_R \times \norm x }} {{axiom | n = \text N 3         | lc= Triangle inequality:         | q = \forall x, y \in V         | ml= \norm {x + y}         | mo= \le         | mr= \norm x + \norm y }} {{end-axiom}}
Definition:Topology Induced by Metric/Definition 1
The '''topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$''' is defined as the set $\tau$ of all open sets of $M$.
Definition:Topology Induced by Metric/Definition 2
The '''topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$''' is defined as the topology $\tau$ generated by the basis consisting of the set of all open $\epsilon$-balls in $M$.
Definition:Closed Set/Real Analysis
=== Real Numbers === {{:Definition:Closed Set/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Closed Set/Real Analysis/Real Euclidean Space}}
Definition:System of Neighborhoods
Let $M = \left({A, d}\right)$ be a metric space. Let $a \in A$. Let $\mathcal N_a$ be the set of all neighborhoods of $a$ in $M$. Then $\mathcal N_a$ is the '''system of neighborhoods''' of the point $a$.
Definition:Continuous Mapping (Metric Space)/Point/Definition 4
$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' : :for each neighborhood $N'$ of $f \left({a}\right)$ in $M_2$ there exists a corresponding neighborhood $N$ of $a$ in $M_1$ such that $f \left[{N}\right] \subseteq N'$.
Definition:Basis for Neighborhood System
Let $M = \left({A, d}\right)$ be a metric space. Let $a \in A$ be a point in $M$. Let $\mathcal B_a$ be a set of neighborhoods of $a$ in $M$. Then $\mathcal B_a$ is a '''basis for the neighborhood system at $a$''' iff: :$\forall N_a \subseteq M: \exists B \in \mathcal B_a: B \subseteq N_a$ where $N_a$ denotes a neighborhood of $a$ in $M$. That is, $\mathcal B_a$ is a '''basis for the neighborhood system at $a$''' iff every neighborhood of $a$ contains an element of $\mathcal B_a$ as a subset.
Definition:Limit of Real Function/Limit at Infinity
=== Limit at (Positive) Infinity === {{:Definition:Limit of Real Function/Limit at Infinity/Positive}} === Limit at Negative Infinity === {{:Definition:Limit of Real Function/Limit at Infinity/Negative}}
Definition:Convergent Mapping/Metric Space
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $f \left({x}\right)$ tend to the limit $L$ as $x$ tends to $c$. Then $f$ '''converges to the limit $L$ as $x$ tends to $c$'''.
Definition:Convergent Mapping/Complex Function
Let $f: \C \to \C$ be a complex function defined everywhere on $\C$ ''except possibly'' at $c$. Let $f \left({z}\right)$ tend to the limit $L$ as $z$ tends to $c$. Then $f$ '''converges to the limit $L$ as $z$ tends to $c$'''.
Definition:Convergent Mapping/Real Function
Let $f: \R \to \R$ be a real function defined everywhere on $A_1$ ''except possibly'' at $c$. Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$. Then $f$ '''converges to the limit $L$ as $x$ tends to $c$'''.
Definition:Convergent Sequence/Metric Space/Definition 1
$\sequence {x_k}$ '''converges to the limit $l \in A$''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$
Definition:Convergent Sequence/Metric Space/Definition 2
$\sequence {x_k}$ '''converges to the limit $l \in A$''' : :$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$ where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.
Definition:Convergent Sequence/Metric Space/Definition 3
$\sequence {x_k}$ '''converges to the limit $l \in A$''' : :$\displaystyle \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$
Definition:Almost All/Set Theory
=== Countable === {{:Definition:Almost All/Set Theory/Countable}} === Uncountable === {{:Definition:Almost All/Set Theory/Uncountable}}
Definition:Limit of Sequence/Real Numbers
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ converge to a value $l \in \R$. Then $l$ is a '''limit of $\sequence {x_n}$ as $n$ tends to infinity'''.
Definition:Convergent Sequence/Real Numbers
Let $\sequence {x_k}$ be a sequence in $\R$. The sequence $\sequence {x_k}$ '''converges to the limit $l \in \R$''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$ where $\size x$ denotes the absolute value of $x$.
Definition:Limit of Sequence/Rational Numbers
Let $\left \langle {x_n} \right \rangle$ be a sequence in $\Q$. Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in \R$, where $\R$ denotes the set of real numbers. Then $l$ is a '''limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity'''.
Definition:Convergent Sequence/Rational Numbers
Let $\sequence {x_k}$ be a sequence in $\Q$. $\sequence {x_k}$ '''converges to the limit $l \in \R$''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$ where $\size x$ is the absolute value of $x$.
Definition:Limit of Sequence/Complex Numbers
Let $\sequence {z_n}$ be a sequence in $\C$. Let $\sequence {z_n}$ converge to a value $l \in \C$. Then $l$ is a '''limit of $\sequence {z_n}$ as $n$ tends to infinity'''.
Definition:Convergent Sequence/Complex Numbers/Definition 1
Let $\sequence {z_k}$ be a sequence in $\C$. $\sequence {z_k}$ '''converges to the limit $c \in \C$''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$ where $\cmod z$ denotes the modulus of $z$.
Definition:Rational Number/Fraction/Vulgar
A '''vulgar fraction''' is a rational number whose absolute value is less than $1$, expressed in the form $r = \dfrac p q$, where $p$ and $q$ are integers.
Definition:Rational Number/Fraction/Improper
An '''improper fraction''' is a rational number whose absolute value is greater than $1$, specifically when expressed in the form $r = \dfrac p q$ where $p$ and $q$ are integers such that $p > q$.
Definition:Rational Number/Fraction/Mixed Number
A '''mixed number''' is a rational number whose absolute value is greater than $1$, expressed in the form $r = n \frac p q$ where: : $p$ and $q$ are integers such that $p < q$ : $r = n + \dfrac p q$
Definition:Absolute Value/Definition 1
Let $x \in \R$ be a real number. The '''absolute value''' of $x$ is denoted $\size x$, and is defined using the usual ordering on the real numbers as follows: :$\size x = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$
Definition:Absolute Value/Definition 2
Let $x \in \R$ be a real number. The '''absolute value''' of $x$ is denoted $\size x$, and is defined as: :$\size x = +\sqrt {x^2}$ where $+\sqrt {x^2}$ is the positive square root of $x^2$.
Definition:Absolute Value/Number Classes
The absolute value function applies to the various number classes as follows: : Natural numbers $\N$: All elements of $\N$ are greater than or equal to zero, so the concept is irrelevant. : Integers $\Z$: As defined here. : Rational numbers $\Q$: As defined here. : Real numbers $\R$: As defined here. : Complex numbers $\C$: As $\C$ is not an ordered set, the definition of the absolute value function based upon whether a complex number is greater than or less than zero cannot be applied. The notation $\cmod z$, where $z \in \C$, is defined as the modulus of $z$ and has a different meaning.
Definition:Open Ball/Center
In $\map {B_\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\epsilon$-ball.
Definition:Basis for Open Sets (Metric Space)
Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal B$ be a set of open sets of $M$. Then $\mathcal B$ is a '''basis for the open sets of $M$''' iff: :for each open set $U$ of $M$, $U$ is the union of sets of $\mathcal B$.
Definition:Closure (Metric Space)
Let $M = \struct {A, d}$ be a metric space. Let $H \subseteq A$. Let $H'$ be the set of limit points of $H$. Let $H^i$ be the set of isolated points of $H$. The '''closure of $H$ (in $M$)''' is the union of all isolated points of $H$ and all limit points of $H$: :$H^- := H' \cup H^i$
Definition:Closure (Topology)/Notation
The topological closure of $H$ is variously denoted: :$\map \cl H$ :$\map {\mathrm {Cl} } H$ :$\overline H$ :$H^-$ Of these, it can be argued that $\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement. $\map \cl H$ and $\map {\mathrm {Cl} } H$ are cumbersome, but they have the advantage of being clear. $H^-$ is neat and compact, but has the disadvantage of being relatively obscure. On , $H^-$ is notation of choice, although $\map \cl H$ can also be found in places.
Definition:Hermitian Conjugate
Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over the complex numbers $\C$. Then the '''Hermitian conjugate''' of $\mathbf A$ is denoted $\mathbf A^\dagger$ and is defined as: :$\mathbf A^\dagger = \sqbrk \beta_{n m}: \forall i \in \set {1, 2, \ldots, n}, j \in \set {1, 2, \ldots, m}: \beta_{i j} = \overline {\alpha_{j i} }$ where $\overline {\alpha_{j i} }$ denotes the complex conjugate of $\alpha_{j i}$.
Definition:Hermitian Matrix
Let $\mathbf A$ be a square matrix over $\C$. $\mathbf A$ is '''Hermitian''' : :$\mathbf A = \mathbf A^\dagger$ where $\mathbf A^\dagger$ is the Hermitian conjugate of $\mathbf A$.
Definition:Binomial Coefficient/Technical Note
The $\LaTeX$ code to render the binomial coefficient $\dbinom n k$ can be written in the following ways: :<code>\dbinom n k</code> or: :<code>\displaystyle {n \choose k}</code> The <code>\dbinom</code> form is preferred on  because it is simpler. It is in fact an abbreviated form of <code>\displaystyle \binom n k</code>, which is the preferred construction when <code>\displaystyle</code> is required for another entity in the expression. While the form <code>\binom n k</code> is valid $\LaTeX$ syntax, it renders the entity in the reduced size inline style: $\binom n k$ which  does not endorse. To render compound arguments, braces are needed to delimit the parameter when using <code>\dbinom</code>, but (confusingly) not <code>\choose</code>. For example, to render $\dbinom {a + b} {c d}$ the following can be used: :<code><nowiki>\dbinom {a + b} {c d}</nowiki></code> or: :<code><nowiki>\displaystyle {a + b \choose c d}</nowiki></code> $\displaystyle {a + b \choose c d}$ Again, for consistency across , the <code>\dbinom</code> form is preferred.
Definition:Unitary Matrix
Let $\mathbf U$ be an invertible square matrix over the complex numbers $\C$. Then $\mathbf U$ is '''unitary''' : :$\mathbf U^{-1} = \mathbf U^\dagger$ where: :$\mathbf U^{-1}$ is the inverse of $\mathbf U$ :$\mathbf U^\dagger$ is the Hermitian conjugate of $\mathbf U$.
Definition:Homeomorphism/Metric Spaces
=== Definition 1 === {{:Definition:Homeomorphism/Metric Spaces/Definition 1}} === Definition 2 === {{:Definition:Homeomorphism/Metric Spaces/Definition 2}} === Definition 3 === {{:Definition:Homeomorphism/Metric Spaces/Definition 3}} === Definition 4 === {{:Definition:Homeomorphism/Metric Spaces/Definition 4}}
Definition:Möbius Transformation
A '''Möbius transformation''' is a mapping $f: \overline \C \to \overline \C$ of the form: :$\map f z = \dfrac {a z + b} {c z + d}$ where: :$\overline C$ denotes the extended complex plane :$a, b, c, d \in \C$ such that $a d - b c \ne 0$ We define: :$\map f {-\dfrac d c} = \infty$ if $c \ne 0$, and: :$\map f \infty = \begin{cases} \dfrac a c & : c \ne 0 \\ \infty & : c = 0 \end{cases}$
Definition:Liouville Number
A real number $x$ is a '''Liouville number''' if for all $n \in \N$, there exist $p, q \in \Z$ (which may depend on $n$) with $q > 1$ such that: :$0 < \size {x - \dfrac p q} < \dfrac 1 {q^n}$
Definition:Bell Number
The '''Bell Numbers''' $B_n$ are a sequence of natural numbers defined as the number of ways a set with $n$ elements can be partitioned.
Definition:Catalan's Constant
'''Catalan's constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = G       | r = \map \beta 2       | c =  }} {{eqn | r = \sum_{n \mathop = 0}^{\infty} \frac{\paren {-1}^n} {\paren {2 n + 1}^2}       | c =  }} {{eqn | r = \frac 1 {1^2} - \frac 1 {3^2} + \frac 1 {5^2} - \frac 1 {7^2} + \cdots       | c =  }} {{end-eqn}} where $\beta$ is the Dirichlet beta function. Its numerical value is approximately: :$G = 0 \cdotp 91596 \, 55941 \, 77219 \, 01505 \, 46035 \, 14932 \, 38411 \, 0774 \ldots$
Definition:Lipschitz Equivalence/Metrics/Definition 1
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $\exists h, k \in \R_{>0}$ such that: : $\forall x, y \in A: h \map {d_2} {x, y} \le \map {d_1} {x, y} \le k \map {d_2} {x, y}$ Then $d_1$ and $d_2$ are described as '''Lipschitz equivalent'''.
Definition:Lipschitz Equivalence/Metrics/Definition 2
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $\exists K_1, K_2 \in \R_{>0}$ such that: : $(1): \quad \forall x, y \in A: \map {d_2} {x, y} \le K_1 \map {d_1} {x, y}$ : $(2): \quad \forall x, y \in A: \map {d_1} {x, y} \le K_2 \map {d_2} {x, y}$ Then $d_1$ and $d_2$ are described as '''Lipschitz equivalent'''.
Definition:Homeomorphism/Metric Spaces/Definition 1
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : $f$ is continuous from $M_1$ to $M_2$ : $f^{-1}$ is continuous from $M_2$ to $M_1$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.
Definition:Homeomorphism/Metric Spaces/Definition 2
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : for all $U \subseteq A_1$, $U$ is an open set of $M_1$  $f \left[{U}\right]$ is an open set of $M_2$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.
Definition:Homeomorphism/Metric Spaces/Definition 3
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : for all $V \subseteq A_1$, $V$ is a closed set of $M_1$  $f \left[{V}\right]$ is a closed set of $M_2$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.
Definition:Homeomorphism/Metric Spaces/Definition 4
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : for all $a \in A_1$ and $N \subseteq A_1$, $N$ is a neighborhood of $a$  $f \left[{N}\right]$ is a neighborhood of $f \left({a}\right)$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.
Definition:Cardinality of Continuum
The '''Cardinality of the Continuum''' is the cardinality of the set of real numbers, denoted $\left \lvert {\R}\right \rvert$ or $\mathfrak c$. {{refactor|State and prove this in a separate page}} It is an infinite cardinal number.
Definition:Exact Sequence of Groups
Let $\left({G, \circ}\right)$ be a group. Consider the sequence of groups $\left\langle{G_i}\right\rangle$ and group homomorphisms $\phi_i$: :$\displaystyle \cdots \stackrel{\phi_{i-2}}{\longrightarrow} G_{i-1} \stackrel{\phi_{i-1}}{\longrightarrow} G_i \stackrel{\phi_i}{\longrightarrow} G_{i+1} \stackrel{\phi_{i+1}}{\longrightarrow} \cdots$ $\left\langle{G_i}\right\rangle$ is  '''exact''' : :$\forall i: \operatorname{Im} \left({\phi_i}\right) = \ker \left({\phi_{i+1} }\right)$ where: :$\operatorname{Im} \left({\phi_i}\right)$ denotes the image of $\phi_i$ :$\ker \left({\phi_{i+1} }\right)$ denotes the kernel of $\phi_{i+1}$.
Definition:Short Exact Sequence of Groups
Let $\left({G, \cdot}\right)$ be a group. An exact sequence of the form :$1 \longrightarrow K \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta}{\longrightarrow} H \longrightarrow 1$ is called a '''short exact sequence''', where $1$ represents the trivial group.
Definition:Dozen
A '''dozen''' is another name for a set whose cardinality is $12$.
Definition:Gross
A '''gross''' is another name for a set whose cardinality is $12 \times 12$, or $144$.
Definition:Score
A '''score''' is another name for a set whose cardinality is $20$.
Definition:Integral Sign
The symbol: :$\displaystyle \int \ldots \rd \mu$ is called the '''integral sign'''. Note that there are two parts to this symbol, which embrace the function $f$ which is being integrated.
Definition:Axis/Positive Direction
Consider a coordinate system whose axes are each aligned with an instance of the real number line $\R$. The direction along an axis in which the corresponding elements of $\R$ are increasing is called the '''positive direction'''.
Definition:Smallest Set by Set Inclusion
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by $\subseteq$ considered as an ordering. Then $T \in \TT$ is the '''smallest set''' of $\TT$  $T$ is the smallest element of $\struct {\TT, \subseteq}$. That is: :$\forall X \in \TT: T \subseteq X$
Definition:Smallest Set by Set Inclusion/Class Theory
Let $A$ be a class. Then a set $m$ is the '''smallest element''' of $A$ (with respect to the inclusion relation) : :$(1): \quad m \in A$ :$(2): \quad \forall S: \paren {S \in A \implies m \subseteq S}$
Definition:Euclidean Space/Euclidean Topology/Real
Let $\R^n$ be an $n$-dimensional real vector space. Let $M = \struct {\R^n, d}$ be a real Euclidean $n$-space. The topology $\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.
Definition:Euclidean Space/Euclidean Topology/Rational
Let $\Q^n$ be an $n$-dimensional vector space of rational numbers. Let $M = \left({\Q^n, d}\right)$ be a rational Euclidean $n$-space. The topology induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.
Definition:Euclidean Space/Euclidean Topology/Complex
Let $\C$ be the complex plane. Let $M = \left({\C, d}\right)$ be a complex Euclidean space. The topology induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.
Definition:Neighborhood Filter/Point
Let $x \in S$. Let $\mathcal N_x$ be the set of all neighborhoods of $x$ in $T$. Then $\mathcal N_x$ is the '''neighborhood filter''' of $x$.
Definition:Separated by Neighborhoods/Sets
{{:Definition:Separated by Neighborhoods/Sets/Neighborhoods}}
Definition:Separated by Neighborhoods/Points
{{:Definition:Separated by Neighborhoods/Points/Neighborhoods}}
Definition:Hausdorff Space/Definition 3
$\left({S, \tau}\right)$ is a '''Hausdorff space''' or '''$T_2$ space''' : :$\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$ That is: : for any two distinct elements $x, y \in S$ there exist disjoint neighborhoods $N_x, N_y \subseteq S$ containing $x$ and $y$ respectively.
Definition:Separated by Closed Neighborhoods/Points
Let $x, y \in S$ such that: :$\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \O$ where $N_x^-$ and $N_y^-$ are the closures in $T$ of $N_x$ and $N_y$ respectively. That is, that $x$ and $y$ both have neighborhoods in $T$ whose closures are disjoint. Then $x$ and $y$ are described as '''separated by closed neighborhoods'''.
Definition:Neighborhood Space
Let $S$ be a set. For each $x \in S$, let there be given a set $\mathcal N_x$ of subsets of $S$ which satisfy the neighborhood space axioms: {{:Axiom:Neighborhood Space Axioms}} The sets $\mathcal N_x$ are the neighborhoods of $x$ in $S$. Let $\mathcal N$ be the set of open sets of $S$: :$\mathcal N = \left\{{U \subseteq S: U}\right.$ is a neighborhood of each of its elements $\left.{}\right\}$ The set $S$ together with $\mathcal N$ is called a '''neighborhood space''' and is denoted $\left({S, \mathcal N}\right)$.
Definition:Neighborhood Filter/Set
Let $A \subseteq S$ such that $A \ne \O$. Let $\NN_A$ be the set of all neighborhoods of $A$. Then $\NN_A$ is the '''neighborhood filter''' of $A$.
Definition:Neighborhood (Neighborhood Space)
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $x \in S$. Let $\mathcal N_x$ satisfy the neighborhood space axioms. Then each element $N$ of $\mathcal N_x$ is called a '''neighborhood''' of $x$.
Definition:Open Set (Neighborhood Space)
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $U \subseteq S$ be a neighborhood of each of its elements. Then $U$ is an '''open set''' of $S$.
Definition:Topological Space Induced by Neighborhood Space
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $\tau$ be the set of open sets of $\left({S, \mathcal N}\right)$. Then $\left({S, \tau}\right)$ is the '''topological space induced by $\left({S, \mathcal N}\right)$'''.
Definition:Combinatorics
'''Combinatorics''' is that branch of mathematics concerned with counting things. '''Combinatorial''' problems are so named because they are exercises in counting the number of combinations of various objects. It has been stated that it is the core of the discipline of discrete mathematics.
Definition:Binomial Coefficient/Notation
The notation $\dbinom n k$ for the binomial coefficient was introduced by {{AuthorRef|Andreas Freiherr von Ettingshausen}} in his $1826$ work {{BookLink|Die kombinatorische Analysis, als Vorbereitungslehre zum Studium der theoretischen höheren Mathematik|Andreas v. Ettingshausen}}. It appears to have become the ''de facto'' standard in recent years. As a result, $\dbinom n k$ is frequently voiced '''the binomial coefficient $n$ over $k$'''. Other notations include: :$C \left({n, k}\right)$ :${}^n C_k$ :${}_n C_k$ :$C^n_k$ :${C_n}^k$ all of which can cause a certain degree of confusion.
Definition:Binomial Coefficient/Integers/Definition 1
Let $n \in \Z_{\ge 0}$ and $k \in \Z$. Then the '''binomial coefficient''' $\dbinom n k$ is defined as: :$\dbinom n k = \begin{cases} \dfrac {n!} {k! \paren {n - k}!} & : 0 \le k \le n \\ & \\ 0 & : \text { otherwise } \end{cases}$ where $n!$ denotes the factorial of $n$.