hcju/proofwikips huggingface dataset:
Definition:Complex Point at Infinity
The zero in the set of complex numbers $\C$ has no inverse for multiplication. That is, the expression: :$\dfrac 1 0$ has no meaning. The '''point at infinity''' is the element added to $\C$ in order to allow $\C$ to be closed under division: :$\forall x, y \in \C: \dfrac x y \in \C$ The set $\C$ with that point added is known as the extended complex plane. Conceptually, it can be imagined as a point which is '''''at infinity''''' in '''''all''''' directions. It can also be considered as the $N$ point on the Riemann sphere which does not map to the complex plane. This point can be denoted $\infty$.
Definition:Definable Truth Function
Let $f: \Bbb B^n \to \Bbb B$ be a truth function. Let $S$ be a set of truth functions. Then $f$ is '''definable from $S$''' iff there exist: :a truth function $g: \Bbb B^m \to \Bbb B$, obtained by composition of truth functions from $S$ :an injection $i: \Bbb B^n \to \Bbb B^m$ such that: :$f = g \circ i$ {{expand|example}}
Definition:Satisfiable/Boolean Interpretations
Let $\mathbf A$ be a WFF of propositional logic. $\mathbf A$ is called '''satisfiable (for boolean interpretations)''' iff: :$v \left({\mathbf A}\right) = T$ for some boolean interpretation $v$ for $\mathbf A$. In terms of validity, this can be rendered: :$v \models_{\mathrm{BI}} \mathbf A$ that is, $\mathbf A$ is valid in the boolean interpretation $v$ of $\mathbf A$.
Definition:Falsifiable/Boolean Interpretations
Let $\mathbf A$ be a WFF of propositional logic. $\mathbf A$ is called '''falsifiable (for boolean interpretations)''' iff: :$v \left({\mathbf A}\right) = F$ for some boolean interpretation $v$ for $\mathbf A$. In terms of validity, this can be rendered: :$v \not\models_{\mathrm{BI}} \mathbf A$ that is, $\mathbf A$ is invalid in the boolean interpretation $v$ of $\mathbf A$.
Definition:Extended Real Number Line/Definition 1
The '''extended real number line''' $\overline \R$ is defined as: :$\overline \R := \R \cup \set {+\infty, -\infty}$ that is, the set of real numbers together with two auxiliary symbols: :$+\infty$, '''positive infinity''' :$-\infty$, '''negative infinity''' such that: :$\forall x \in \R: x < +\infty$ :$\forall x \in \R: -\infty < x$
Definition:Extended Real Number Line/Definition 2
The '''extended real number line''' $\overline \R$ is the order completion of the set of real numbers $\R$. The greatest element of $\overline \R$ is often denoted by $+\infty$ and its least element by $-\infty$.
Definition:Extended Complex Plane
The '''extended complex plane''' $\overline \C$ is defined as: :$\overline \C := \C \cup \set \infty$ that is, the set of complex numbers together with the point at infinity.
Definition:Stereographic Projection
Let $\PP$ be a the plane. Let $\mathbb S$ be a sphere which is tangent to $\PP$ at the origin $\tuple {0, 0}$. Let the diameter of $\mathbb S$ perpendicular to $\PP$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\mathbb S$ and $S$ be referred to as the '''south pole''' of $\mathbb S$. Let $A$ be a point on $P$. Let the line $NA$ be constructed. :900px Then $NA$ passes through a point of $\mathbb S$. Thus any point on $P$ can be represented by a point on $\mathbb S$. With this construction, the point $N$ on $\mathbb S$ maps to no point on $\mathbb S$.
Definition:Dot Product/Complex/Definition 2
The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 = \cmod {z_1} \, \cmod{z_2} \cos \theta$ where: :$\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\theta$ denotes the angle between $z_1$ and $z_2$.
Definition:Dot Product/Complex/Definition 1
The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 = x_1 x_2 + y_1 y_2$
Definition:Dot Product/Complex
Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers. === Definition 1 === {{:Definition:Dot Product/Complex/Definition 1}} === Definition 2 === {{:Definition:Dot Product/Complex/Definition 2}} === Definition 3 === {{:Definition:Dot Product/Complex/Definition 3}} === Definition 4 === {{:Definition:Dot Product/Complex/Definition 4}}
Definition:Dot Product/Complex/Definition 3
The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 := \map \Re {\overline {z_1} z_2}$ where: :$\map \Re z$ denotes the real part of a complex number $z$ :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.
Definition:Dot Product/Complex/Definition 4
The '''dot product''' of $z_1$ and $z_2$ is defined as: :$z_1 \circ z_2 := \dfrac {\overline {z_1} z_2 + z_1 \overline {z_2} } 2$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.
Definition:Vector Cross Product/Definition 1
The '''vector cross product''', denoted $\mathbf a \times \mathbf b$, is defined as: :$\mathbf a \times \mathbf b = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k\\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$ where $\begin{vmatrix} \ldots \end{vmatrix}$ is interpreted as a determinant. More directly: : $\mathbf a \times \mathbf b = \mathbf i \paren {a_j b_k - a_k b_j} - \mathbf j \paren {a_i b_k - a_k b_i} + \mathbf k \paren {a_i b_j - a_j b_i}$
Definition:Vector Cross Product/Definition 2
The '''vector cross product''', denoted $\mathbf a \times \mathbf b$, is defined as: :$\mathbf a \times \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \sin \theta \hat {\mathbf n}$ where: :$\norm {\mathbf a}$ denotes the length of $\mathbf a$ :$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction :$\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right hand rule.
Definition:Orthogonal (Linear Algebra)/Set
Let $S = \set {u_1, \ldots, u_n}$ be a subset of $V$. Then $S$ is an '''orthogonal set'''  its elements are pairwise orthogonal: :$\forall i \ne j: \innerprod {u_i}, {u_j} = 0$
Definition:Vector Cross Product/Complex
Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers. === Definition 1 === {{:Definition:Vector Cross Product/Complex/Definition 1}} === Definition 2 === {{:Definition:Vector Cross Product/Complex/Definition 2}} === Definition 3 === {{:Definition:Vector Cross Product/Complex/Definition 3}} === Definition 4 === {{:Definition:Vector Cross Product/Complex/Definition 4}}
Definition:Vector Cross Product/Complex/Definition 2
The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$ where: :$\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.
Definition:Vector Cross Product/Complex/Definition 1
The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 = x_1 y_2 - y_1 x_2$
Definition:Vector Cross Product/Complex/Definition 3
The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 := \map \Im {\overline {z_1} z_2}$ where: :$\map \Im z$ denotes the imaginary part of a complex number $z$ :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.
Definition:Vector Cross Product/Complex/Definition 4
The '''cross product''' of $z_1$ and $z_2$ is defined as: :$z_1 \times z_2 := \dfrac {\overline {z_1} z_2 - z_1 \overline {z_2}} {2 i}$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\overline {z_1} z_2$ denotes complex multiplication.
Definition:Bound of Real-Valued Function
Let $S$ be a set. Let $f: S \to \R$ be a real-valued function. Let $f$ be bounded. Then $B$ is a '''bound''' for $f$ : :$\forall x \in S: B \ge \size {\map f x}$
Definition:Bounded Ordered Set/Unbounded
Let $\left({S, \preceq}\right)$ be an ordered set. A subset $T \subseteq S$ is '''unbounded (in $S$)'''  it is not bounded.
Definition:Bounded Above Set/Unbounded
Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''unbounded above (in $S$)'''  it is not bounded above.
Definition:Bounded Below Set/Unbounded
Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''unbounded below (in $S$)'''  it is not bounded below.
Definition:Bounded Mapping/Unbounded
$f$ is '''unbounded'''  it is neither bounded above or bounded below.
Definition:Bounded Sequence/Unbounded
A sequence which is not bounded is '''unbounded'''.
Definition:Decision Procedure/Refutation Procedure
Given a decision procedure for satisfiability, one can craft a decision procedure for tautologies in the following way: Suppose one wanted to decide if a propositional formula $\mathbf A$ is a tautology. Then apply the given procedure to decide if its negation $\neg \mathbf A$ is satisfiable. Now: :If $\neg \mathbf A$ is not satisfiable, then by Tautology iff Negation is Unsatisfiable, $\mathbf A$ is a tautology. :If $\neg \mathbf A$ is satisfiable, then by Satisfiable iff Negation is Falsifiable, $\mathbf A$ is falsifiable, so cannot be a tautology. Hence we have crafted a decision procedure for tautologies. Such a procedure is called a '''refutation procedure''', because it proceeds by refuting, i.e. proving unsatisfiability of the negation of the formula at hand.
Definition:Bounded Below Mapping/Unbounded
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is '''unbounded below (in $T \ $)'''  there exists no $L \in S$ such that: :$\forall x \in S: L \preceq \map f x$
Definition:Upper Bound of Sequence
A special case of an upper bound of a mapping is an '''upper bound of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\sequence {x_n}$ be bounded above in $T$ by $H \in T$. Then $H$ is an '''upper bound of $\sequence {x_n}$'''.
Definition:Lower Bound of Sequence
A special case of a lower bound of a mapping is a '''lower bound of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\sequence {x_n}$ be bounded below in $T$ by $L \in T$. Then $L$ is a '''lower bound of $\sequence {x_n}$'''.
Definition:Upper Bound of Number
When considering the upper bound of a set of numbers, it is commonplace to ignore the set and instead refer just to the number itself. Thus the construction: :''The set of numbers which fulfil the propositional function $P \left({n}\right)$ is bounded above with the upper bound $N$'' would be reported as: :''The number $n$ such that $P \left({n}\right)$ has the upper bound $N$''. This construct obscures the details of what is actually being stated. Its use on  is considered an abuse of notation and so discouraged. This also applies in the case where it is the upper bound of a mapping which is under discussion.
Definition:Infimum of Mapping/Real-Valued Function
{{:Definition:Infimum of Mapping/Real-Valued Function/Definition 1}}
Definition:Bounded Mapping/Real-Valued/Definition 1
$f$ is '''bounded on $S$''' : :$f$ is bounded above on $S$ and also: :$f$ is bounded below on $S$.
Definition:Bounded Mapping/Real-Valued/Definition 2
$f$ is '''bounded on $S$''' : :$\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$ where $\size {\map f x}$ denotes the absolute value of $\map f x$.
Definition:Bounded Above Mapping/Real-Valued
Let $f: S \to \R$ be a real-valued function. $f$ is '''bounded above on $S$''' by the upper bound $H$ : :$\forall x \in S: \map f x \le H$
Definition:Bounded Above Mapping/Real-Valued/Unbounded
Let $f: S \to \R$ be a real-valued function. Then $f$ is '''unbounded above on $S$'''  it is not bounded above on $S$: :$\neg \exists H \in \R: \forall x \in S: \map f x \le H$
Definition:Bounded Above Mapping/Unbounded
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is '''unbounded above on $S$'''  it is not bounded above on $S$: :$\neg \exists H \in T: \forall x \in S: \map f x \preceq H$
Definition:Bounded Below Mapping/Real-Valued
Let $f: S \to \R$ be a real-valued function. Then $f$ is '''bounded below on $S$''' by the lower bound $L$ : :$\forall x \in S: L \le \map f x$
Definition:Bounded Below Mapping/Real-Valued/Unbounded Below
Let $f: S \to \R$ be a real-valued function. Then $f$ is '''unbounded below on $S$'''  it is not bounded below on $S$: :$\neg \exists L \in \R: \forall x \in S: L \le \map f x$
Definition:Bound of Subset of Real Numbers
Let $S \subseteq \R$ be a subset of the real numbers $\R$. Let $S$ be bounded. Then $K$ is a '''bound''' for $S$ : :$\forall x \in S: \size x \le K$
Definition:Bounded Ordered Set/Real Numbers/Definition 1
Let $T \subseteq \R$ be both bounded below and bounded above in $\R$. Then $T$ is '''bounded in $\R$'''.
Definition:Bounded Ordered Set/Real Numbers/Definition 2
Let $T \subseteq \R$ be a subset of $\R$ such that: :$\exists K \in \R: \forall x \in T: \size x \le K$ where $\size x$ denotes the absolute value of $x$. Then $T$ is '''bounded in $\R$'''.
Definition:Bounded Ordered Set/Real Numbers
{{:Definition:Bounded Ordered Set/Real Numbers/Definition 1}}
Definition:Upper Bound of Set/Real Numbers
Let $\R$ be the set of real numbers. Let $T$ be a subset of $\R$. An '''upper bound for $T$ (in $\R$)''' is an element $M \in \R$ such that: :$\forall t \in T: t \le M$ That is, $M$ is greater than or equal to every element of $T$.
Definition:Bounded Above Set/Real Numbers
Let $\R$ be the set of real numbers. A subset $T \subseteq \R$ is '''bounded above (in $\R$)'''  $T$ admits an upper bound (in $\R$).
Definition:Bounded Below Set/Real Numbers
Let $\R$ be the set of real numbers. A subset $T \subseteq \R$ is '''bounded below (in $\R$)'''  $T$ admits a lower bound (in $\R$).
Definition:Lower Bound of Set/Real Numbers
Let $\R$ be the set of real numbers. Let $T$ be a subset of $S$. A '''lower bound for $T$ (in $\R$)''' is an element $m \in \R$ such that: :$\forall t \in T: m \le t$
Definition:Bounded Below Mapping/Real-Valued/Unbounded
Let $f: S \to \R$ be a real-valued function. Then $f$ is '''unbounded below on $S$'''  it is not bounded below on $S$: :$\neg \exists L \in \R: \forall x \in S: L \le \map f x$
Definition:Upper Bound of Mapping/Real-Valued
Let $f: S \to \R$ be a real-valued function. Let $f$ be bounded above in $\R$ by $H \in \R$. Then $H$ is an '''upper bound of $f$'''.
Definition:Lower Bound of Mapping/Real-Valued
Let $f: S \to \R$ be a real-valued function. Let $f$ be bounded below in $T$ by $L \in T$. Then $L$ is a '''lower bound of $f$'''.
Definition:Supremum of Mapping/Real-Valued Function/Definition 1
The '''supremum of $f$ on $S$''' is defined by: :$\displaystyle \sup_{x \mathop \in S} \map f x := \sup f \sqbrk S$ where :$\sup f \sqbrk S$ is the supremum in $\R$ of the image of $S$ under $f$.
Definition:Supremum of Mapping/Real-Valued Function/Definition 2
The '''supremum of $f$ on $S$''' is defined as $\displaystyle \sup_{x \mathop \in S} \map f x := K \in \R$ such that: :$(1): \quad \forall x \in S: \map f x \le K$ :$(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: \map f x > K - \epsilon$
Definition:Infimum of Mapping/Real-Valued Function/Definition 1
The '''infimum of $f$ on $S$''' is defined by: :$\displaystyle \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$ where :$\inf f \sqbrk S$ is the infimum in $\R$ of the image of $S$ under $f$.
Definition:Infimum of Mapping/Real-Valued Function/Definition 2
The '''infimum of $f$ on $S$''' is defined as $\displaystyle \inf_{x \mathop \in S} \map f x := k \in \R$ such that: :$(1): \quad \forall x \in S: k \le \map f x$ :$(2): \quad \forall \epsilon \in \R_{>0}: \exists x \in S: \map f x < k + \epsilon$
Definition:Bounded Above Sequence/Real
Let $\sequence {x_n}$ be a real sequence. Then $\sequence {x_n}$ is '''bounded above''' : :$\exists M \in \R: \forall i \in \N: x_i \le M$
Definition:Bounded Below Sequence/Real
Let $\sequence {x_n}$ be a real sequence. Then $\sequence {x_n}$ is '''bounded below''' : : $\exists m \in \R: \forall i \in \N: m \le x_i$
Definition:Bounded Above Sequence/Real/Unbounded
$\sequence {x_n}$ is '''unbounded above'''  there exists no $M$ in $\R$ such that: :$\forall i \in \N: x_i \le M$
Definition:Bounded Below Sequence/Real/Unbounded
$\sequence {x_n}$ is '''unbounded below'''  there exists no $m$ in $\R$ such that: :$\forall i \in \N: m \le x_i$
Definition:Bounded Sequence/Real
Let $\sequence {x_n}$ be a real sequence. Then $\sequence {x_n}$ is '''bounded'''  $\exists m, M \in \R$ such that $\forall i \in \N$: :$m \le x_i$ :$x_i \le M$
Definition:Bounded Sequence/Real/Unbounded
$\sequence {x_n}$ is '''unbounded'''  it is not bounded.
Definition:Bounded Mapping/Real-Valued/Unbounded
$f$ is '''unbounded'''  it is neither bounded above nor bounded below.
Definition:Bounded Ordered Set/Real Numbers/Unbounded
$T \subseteq \R$ is '''unbounded (in $\R$)'''  it is not bounded.
Definition:Bounded Above Set/Real Numbers/Unbounded
$T \subseteq \R$ is '''unbounded above (in $\R$)'''  it is not bounded above.
Definition:Bounded Below Set/Real Numbers/Unbounded
$T \subseteq \R$ is '''unbounded below (in $\R$)'''  it is not bounded below.
Definition:Lower Bound of Sequence/Real
Let $\sequence {x_n}$ be a real sequence. Let $\sequence {x_n}$ be bounded below in $T$ by $L \in \R$. Then $L$ is a '''lower bound of $\sequence {x_n}$'''.
Definition:Upper Bound of Sequence/Real
Let $\sequence {x_n}$ be a real sequence. Let $\sequence {x_n}$ be bounded above by $H \in \R$. Then $H$ is an '''upper bound of $\sequence {x_n}$'''.
Definition:Infimum of Set/Real Numbers
Let $T \subseteq \R$. A real number $c \in \R$ is the '''infimum of $T$ in $\R$''' : :$(1): \quad c$ is a lower bound of $T$ in $\R$ :$(2): \quad d \le c$ for all lower bounds $d$ of $T$ in $\R$.
Definition:Supremum of Set/Real Numbers
Let $T \subseteq \R$ be a subset of the real numbers. A real number $c \in \R$ is the '''supremum of $T$ in $\R$''' : :$(1): \quad c$ is an upper bound of $T$ in $\R$ :$(2): \quad c \le d$ for all upper bounds $d$ of $T$ in $\R$.
Definition:Bounded Metric Space/Complex
Let $D$ be a subset of the complex plane $\C$. Then '''$D$ is bounded (in $\C$)'''  there exists $M \in \R$ such that: : $\forall z \in D: \cmod z \le M$
Definition:Bounded Metric Space/Complex/Unbounded
Let $D$ be a subset of the complex plane $\C$. Then '''$D$ is unbounded (in $\C$)''' : : $\nexists M \in \R: \forall z \in D: \cmod z \le M$ That is, if $D$ is not bounded in $\C$.
Definition:Compact Space/Metric Space/Complex
Let $D$ be a subset of the complex plane $\C$. Then $D$ is '''compact (in $\C$)''' : : $D$ is closed in $\C$ and : $D$ is bounded in $\C$.
Definition:Interior Point (Topology)/Definition 1
Let $h \in H$. Then $h$ is an '''interior point''' of $H$ : :$h \in H^\circ$ where $H^\circ$ denotes the interior of $H$.
Definition:Interior Point (Topology)/Definition 2
Let $h \in H$. $h$ is an '''interior point''' of $H$  $h$ has an open neighborhood $N_h$ such that $N_h \subseteq H$.
Definition:Theory
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a set of $\mathcal L$-formulas. Then $\mathcal F$ is an '''$\mathcal L$-theory''' iff, for every $\phi \in \mathcal L$: :$\mathcal F \models_{\mathscr M} \phi \implies \phi \in \mathcal F$ where $\models_{\mathscr M}$ denotes $\mathscr M$-semantic consequence. === Theory of Set of Formulas === {{:Definition:Theory/Set of Formulas}}
Definition:Axiomatization
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathcal L$-theory. An '''axiomatization of $\mathcal F$''' is a subset $\mathcal A \subseteq \mathcal F$ such that: :$\mathcal F = \left\{{\phi \in \mathcal L: \mathcal A \models_{\mathscr M} \phi}\right\}$ That is, all of $\mathcal F$ is a semantic consequence of $\mathcal A$. === Axiom === {{:Definition:Axiomatization/Axiom}}
Definition:Axiomatization/Axiom
Let $\mathcal A$ be an axiomatization of $\mathcal F$. Then a formula $\phi \in \mathcal A$ is called an '''axiom'''.
Definition:Finitely Axiomatizable
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathcal L$-theory. Then $\mathcal F$ is called '''finitely axiomatizable''' iff there exists a finite axiomatization $\mathcal A$ of $\mathcal F$.
Definition:Closed Set/Metric Space/Definition 1
'''$H$ is closed (in $M$)'''  its complement $A \setminus H$ is open in $M$.
Definition:Closed Set/Metric Space/Definition 2
'''$H$ is closed (in $M$)'''  every limit point of $H$ is also a point of $H$.
Definition:Closed Set/Topology
{{:Definition:Closed Set/Topology/Definition 1}}
Definition:Literal/Positive
A '''positive literal''' is an atom $p$ of propositional logic.
Definition:Literal/Negative
A '''negative literal''' is the negation $\neg p$ of an atom $p$ of propositional logic.
Definition:Closed Set/Complex Analysis
Let $S \subseteq \C$ be a subset of the complex plane. '''$S$ is closed (in $\C$)'''  every limit point of $S$ is also a point of $S$. That is:  $S$ contains all its limit points.
Definition:Interior Point (Complex Analysis)
Let $S \subseteq \C$ be a subset of the complex plane. Let $z \in S$. $z$ is an '''interior point''' of $S$  $z$ has an $\epsilon$-neighborhood $\map {N_\epsilon} z$ such that $\map {N_\epsilon} z \subseteq S$.
Definition:Boundary Point (Complex Analysis)
Let $S \subseteq \C$ be a subset of the complex plane. Let $z_0 \in \C$. $z_0$ is a '''boundary point''' of $S$  every $\epsilon$-neighborhood $\map {N_\epsilon} {z_0}$ of $z_0$ contains points of $\C$ in $S$ and also points of $\C$ which are not in $S$.
Definition:Exterior Point (Complex Analysis)/Definition 1
$z_0$ is an '''exterior point''' of $S$  $z_0$ has an $\epsilon$-neighborhood which is disjoint from $S$.
Definition:Exterior Point (Complex Analysis)/Definition 2
$z_0$ is an '''exterior point''' of $S$ : :$z_0$ is not an interior point of $S$ and: :$z_0$ is not a boundary point of $S$.
Definition:Exterior Point (Complex Analysis)
Let $S \subseteq \C$ be a subset of the complex plane. Let $z_0 \in \C$. === Definition 1 === {{:Definition:Exterior Point (Complex Analysis)/Definition 1}} === Definition 2 === {{:Definition:Exterior Point (Complex Analysis)/Definition 2}}
Definition:Open Set/Complex Analysis/Definition 1
Let $S \subseteq \C$ be a subset of the set of complex numbers. Let: : $\forall z_0 \in S: \exists \epsilon \in \R_{>0}: N_{\epsilon} \left({z_0}\right) \subseteq S$ where $N_{\epsilon} \left({z_0}\right)$ is the $\epsilon$-neighborhood of $z_0$ for $\epsilon$. Then $S$ is an '''open set (of $\C$)''', or '''open (in $\C$)'''.
Definition:Open Set/Complex Analysis/Definition 2
Let $S \subseteq \C$ be a subset of the set of complex numbers. Then $S$ is an '''open set (of $\C$)''', or '''open (in $\C$)'''  every point of $S$ is an interior point.
Definition:Region/Plane
A point set $R$ in the plane is a '''region''' : :$(1): \quad$ Each point of $R$ is the center of a circle all of whose elements consist of points of $R$ :$(2): \quad$ Each point of $R$ can be joined by a curve consisting entirely of points of $R$.
Definition:Interior (Complex Analysis)
Let $H \subseteq \C$ be a subset of the complex plane. The '''interior of $H$''' is the subset of $H$ which consists of the interior points of $H$.
Definition:Connected Set (Complex Analysis)
Let $D \subseteq \C$ be a subset of the set of complex numbers. === Definition 1 === {{:Definition:Connected Set (Complex Analysis)/Definition 1}} === Definition 2 === {{:Definition:Connected Set (Complex Analysis)/Definition 2}}
Definition:Open Region/Complex
Let $D \subseteq \C$ be a subset of the set of complex numbers. $D$ is an '''open region'''  $D$ is: :$(1): \quad$ An open set and :$(2): \quad$ Connected.
Definition:Open Region/Plane
An '''open region''' is a region without its boundary, i.e. the  interior of such a region.
Definition:Closed Region
=== Complex Analysis === {{:Definition:Closed Region/Complex}} === Closed Region in the Plane === {{:Definition:Closed Region/Plane}}
Definition:Closed Region/Plane
A '''closed region''' is a region complete with its boundary.