hcju/proofwikips huggingface dataset:
Definition:Improper Integral
An '''improper integral''' is a definite integral over an interval which is not closed, that is, open or half open, and whose limits of integration are the end points of that interval. When the end point is not actually in the interval, the conventional definition of the definite integral is not valid.
Definition:Lebesgue Measure
Let $\JJ_{ho}^n$ be the set of half-open $n$-rectangles. Let $\map \BB {\R^n}$ be the Borel $\sigma$-algebra on $\R^n$. Let $\lambda^n$ be the $n$-dimensional Lebesgue pre-measure on $\JJ_{ho}^n$. Any measure $\mu$ extending $\lambda^n$ to $\map \BB {\R^n}$ is called '''$n$-dimensional Lebesgue measure'''.
Definition:Measurable Set
Let $\struct {X, \Sigma}$ be a measurable space. A subset $S \subseteq X$ is said to be '''($\Sigma$-)measurable'''  $S \in \Sigma$. === Measurable Set of an Arbitrary Outer Measure === {{:Definition:Measurable Set/Arbitrary Outer Measure}} === Measurable Subset of the Reals === {{:Definition:Measurable Set/Subset of Reals}} === Measurable Subset of $\R^n$ === {{:Definition:Measurable Set/Subsets of Real Space}}
Definition:Strictly Negative
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$. Then $x \in R$ is '''strictly negative''' : :$x \le 0_R$ and $x \ne 0_R$ That $x$ is strictly negative may be (more conveniently) denoted $0_R < x$ or $x > 0_R$. Thus, the set of all strictly negative elements of $R$ is denoted: :$R_{< 0_R} := \set {x \in R: x < 0_R}$
Definition:Algebra of Sets
{{:Definition:Algebra of Sets/Definition 1}}
Definition:Quotient Topology/Quotient Space
The '''quotient space of $S$ by $\RR$''' is the topological space whose points are elements of the quotient set of $\RR$ and whose topology is $\tau_\RR$: :$T_\RR := \struct {S / \RR, \tau_\RR}$
Definition:Torus (Topology)
The '''$n$-dimensional torus''' (or '''$n$-torus)''' $\Bbb T^n$ is defined as the space whose points are those of the cross product of $n$ circles: :$\Bbb T^n = \underbrace{\Bbb S^1 \times \Bbb S^1 \times \ldots \times \Bbb S^1}_{n \text{ times}}$ and whose topology $\tau_{\Bbb T^n}$ is defined as: :$U \in \tau_{\Bbb T^n} \iff \exists U_1, U_2, \ldots, U_n \in \tau_{\Bbb S^1} : U = U_1 \times U_2 \times \ldots \times U_n$ where $\tau_{\Bbb S^1}$ is the topology of the circle. {{explain|In this context, what is the "topology of the circle"?}}
Definition:Extended Real Number Line
=== Definition 1 === {{:Definition:Extended Real Number Line/Definition 1}} === Definition 2 === {{:Definition:Extended Real Number Line/Definition 2}}
Definition:Characteristic Function (Set Theory)
=== Set === {{:Definition:Characteristic Function (Set Theory)/Set}} === Relation === {{:Definition:Characteristic Function (Set Theory)/Relation}}
Definition:Simple Function
Let $\left({X, \Sigma}\right)$ be a measurable space. A real-valued function $f: X \to \R$ is said to be a '''simple function'''  it is a finite linear combination of characteristic functions: :$\displaystyle f = \sum_{k \mathop = 1}^n a_k \chi_{S_k}$ where $a_1, a_2, \ldots, a_n$ are real numbers and each of the sets $S_k$ is $\Sigma$-measurable.
Definition:Lebesgue Integral
Let $\lambda^n$ be a Lebesgue measure on $\R^n$. Let $f: \R^n \to \overline \R$ be a Lebesgue integrable function. Then the $\lambda^n$-integral of $f$: :$\displaystyle \int f \rd \lambda^n$ is called the '''Lebesgue integral of $f$'''.
Definition:Skolem Sequence
A '''Skolem sequence of order $n$''' is a (finite) sequence $S = \left\{{s_1, s_2, \ldots, s_{2 n}}\right\}$ of $2 n$ integers for which: :$(1): \quad$ For every $k \in \left\{{1, 2, \ldots, n}\right\}$ there exist exactly two elements $s_i, s_j \in S$ so that $s_i = s_j = k$ and: :$(2): \quad$ If $s_i = s_j = k$ and $i < j$ then $j - i = k$. {{NamedforDef|Thoralf Albert Skolem|cat = Skolem}}
Definition:Counting Measure
Let $\struct {X, \Sigma}$ be a measurable space. The '''counting measure (on $X$)''', denoted $\size {\, \cdot \,}$, is the measure defined by: :$\size {\, \cdot \,}: \Sigma \to \overline \R, \ \size E := \begin {cases} \map \# E & : \text {$E$ is finite} \\ +\infty & : \text {$E$ is infinite} \end{cases}$ where $\overline \R$ denotes the extended real numbers, and $\#$ denotes cardinality. That $\size {\, \cdot \,}$ is actually a measure is shown on Counting Measure is Measure.
Definition:Standard Discrete Metric
The '''standard discrete metric''' on a set $S$ is the metric satisfying: :$\map d {x, y} = \begin{cases} 0 & : x = y  \\  1 & : x \ne y \end{cases}$ This can be expressed using the Kronecker delta notation as: :$\map d {x, y} = 1 - \delta_{x y}$ The resulting metric space $M = \struct {S, d}$ is '''the standard discrete metric space on $S$'''.
Definition:Euclidean Metric/Real Vector Space
The '''Euclidean metric''' on $\R^n$ is defined as: :$\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.
Definition:Connected Sum
The '''connected sum''' of two manifolds $A^n, B^n$ of dimension $n$ is defined as follows: Let $\Bbb D^n$ be a closed n-disk. Let $\alpha: \Bbb D^n \to A^n$ be a continuous (or, in the case of smooth manifolds, a smooth) injection. Let $\beta: \Bbb D^n \to B^n$ be a similar function.   Define the set: :$S = \paren {A^n \setminus \map \alpha {\paren {\Bbb D^n}^\circ} } \cup \paren {B^n \setminus \map \beta {\paren {\Bbb D^n}^\circ} }$ where: :$\setminus$ denotes set difference :$\paren {\Bbb D^n}^\circ$ denotes the interior of $B^n$. Define an equivalence relation $\sim$ on $S$ as: :$x \sim y \iff \paren {\paren {x = y} \lor \paren {\map {\alpha^{-1} } x = \map {\beta^{-1} } y} }$ Since the interiors of the disks were removed from the manifolds, it necessarily follows that: :$\map {\alpha^{-1} } x, \map {\beta^{-1} } y \in \partial \Bbb D^n$ The '''connected sum''' $A^n \# B^n$ is defined as the quotient space of $S$ under $\sim$.
Definition:Sphere
=== Geometry === {{:Definition:Sphere/Geometry}} === Topology === {{:Definition:Sphere/Topology}} === Metric Space === {{:Definition:Sphere/Metric Space}} === Normed Division Ring === {{:Definition:Sphere/Normed Division Ring}} === Normed Vector Space === {{:Definition:Sphere/Normed Vector Space}} === P-adic Numbers === The definition of an '''sphere''' in the context of the $p$-adic numbers is a direct application of the definition of an sphere in a normed division ring: {{:Definition:Sphere/P-adic Numbers}}
Definition:Argument of Complex Number
Let $z = x + i y$ be a complex number. An '''argument of $z$''', or $\arg z$, is formally defined as a solution to the pair of equations: :$(1): \quad \dfrac x {\cmod z} = \map \cos {\arg z}$ :$(2): \quad \dfrac y {\cmod z} = \map \sin {\arg z}$ where $\cmod z$ is the modulus of $z$. From Sine and Cosine are Periodic on Reals, it follows that if $\theta$ is an '''argument''' of $z$, then so is $\theta + 2 k \pi$ where $k \in \Z$ is ''any'' integer.
Definition:Cantor Set
=== As a Limit of Intersections === {{:Definition:Cantor Set/Limit of Intersections}} === From Ternary Representation === {{:Definition:Cantor Set/Ternary Representation}} === As a Limit of a Decreasing Sequence === {{:Definition:Cantor Set/Limit of Decreasing Sequence}} These definitions are all equivalent, as shown on Equivalence of Definitions of Cantor Set.
Definition:Dihedral Group
The '''dihedral group''' $D_n$ of order $2 n$ is the group of symmetries of the regular $n$-gon.
Definition:Euler-Mascheroni Constant
The '''Euler-Mascheroni constant''' $\gamma$ is the real number that is defined as: {{begin-eqn}} {{eqn | l = \gamma       | o = :=       | r = \lim_{n \mathop \to +\infty} \paren {\sum_{k \mathop = 1}^n \frac 1 k - \int_1^n \frac 1 x \rd x} }} {{eqn | r = \lim_{n \mathop \to +\infty} \paren {H_n - \ln n} }} {{end-eqn}} where $H_n$ is the harmonic series and $\ln$ is the natural logarithm.
Definition:Alternating Group
Let $S_n$ denote the symmetric group on $n$ letters. For any $\pi \in S_n$, let $\map \sgn \pi$ be the sign of $\pi$. The kernel of the mapping $\sgn: S_n \to C_2$ is called the '''alternating group on $n$ letters''' and denoted $A_n$.
Definition:Dicyclic Group
For even $n$, the '''dicyclic group''' $\Dic n$ of order $4 n$ is the group having the presentation: :$\Dic n = \gen {x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$
Definition:Quaternion Group
The dicyclic group $\Dic 2$ is known as the '''quaternion group'''. The elements of $\Dic 2$ are: :$\Dic 2 = \set {e, a, a^2, a^3, b, a b, a^2 b, a^3 b}$
Definition:Power Series
=== Real Domain === {{:Definition:Power Series/Real Domain}} === Complex Domain === {{:Definition:Power Series/Complex Domain}}
Definition:Interval of Convergence
Let $\xi \in \R$ be a real number. Let $\displaystyle \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series in $x$ about $\xi$. Then the set of values of $x$ for which $\map S x$ converges is called the '''interval of convergence''' of $S$.
Definition:Radius of Convergence
=== Real Domain === {{:Definition:Radius of Convergence/Real Domain}} === Complex Domain === {{:Definition:Radius of Convergence/Complex Domain}}
Definition:Conditionally Convergent Series
A series which is convergent, but not absolutely convergent, is said to be '''conditionally convergent'''.
Definition:Divergent (Analysis)
=== Divergent Sequence === {{:Definition:Divergent Sequence}} === Divergent Series === {{:Definition:Divergent Series}} === Divergent Function === {{:Definition:Divergent Function}} === Divergent Improper Integral === {{:Definition:Divergent Improper Integral}}
Definition:Taylor Series
Let $f$ be a real function which is smooth on the open interval $\openint a b$. Let $\xi \in \openint a b$. Then the '''Taylor series expansion of $f$''' about the point $\xi$ is: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ It is ''not'' necessarily the case that this power series is convergent with sum $\map f x$.
Definition:Analytic Function
=== Real Analytic Function === {{:Definition:Analytic Function/Real Numbers}} === Complex Analytic Function === {{:Definition:Analytic Function/Complex Plane}}
Definition:Sine
=== Definition from Triangle === {{:Definition:Sine/Definition from Triangle}} === Definition from Circle === The sine of an angle in a right triangle can be extended to the full circle as follows: {{:Definition:Sine/Definition from Circle}} === Real Numbers === {{:Definition:Sine/Real Function}} === Complex Numbers === {{:Definition:Sine/Complex Function}}
Definition:Cosine
=== Definition from Triangle === {{:Definition:Cosine/Definition from Triangle}} === Definition from Circle === {{:Definition:Cosine/Definition from Circle}} === Real Numbers === {{:Definition:Cosine/Real Function}} === Complex Numbers === {{:Definition:Cosine/Complex Function}}
Definition:Trigonometric Function
There are six basic '''trigonometric functions''': sine, cosine, tangent, cotangent, secant, and cosecant.
Definition:Secant Function
=== Definition from Triangle === {{:Definition:Secant Function/Definition from Triangle}} === Definition from Circle === {{:Definition:Secant Function/Definition from Circle}} === Real Function === {{:Definition:Secant Function/Real}} === Complex Function === {{:Definition:Secant Function/Complex}}
Definition:Cosecant
=== Definition from Triangle === {{:Definition:Cosecant/Definition from Triangle}} === Definition from Circle === {{:Definition:Cosecant/Definition from Circle}} === Real Function === {{:Definition:Cosecant/Real Function}} === Complex Function === {{:Definition:Cosecant/Complex Function}}
Definition:Cotangent
=== Definition from Triangle === {{:Definition:Cotangent/Definition from Triangle}} === Definition from Circle === {{:Definition:Cotangent/Definition from Circle}} === Real Function === {{:Definition:Cotangent/Real Function}} === Complex Function === {{:Definition:Cotangent/Complex Function}}
Definition:Holomorphic Function
=== Complex Function === {{:Definition:Holomorphic Function/Complex Plane}} === Vector-Valued Function ===
Definition:Complex Function
A '''complex function''' is a function whose domain and codomain are subsets of the set of complex numbers $\C$.
Definition:Periodic Function
=== Periodic Real Function === {{:Definition:Periodic Function/Real}} === Periodic Complex Function === {{:Definition:Periodic Function/Complex}}
Definition:Pi
The real number $\pi$ (pronounced '''pie''') is an irrational number (see proof here) whose value is approximately $3.14159\ 26535\ 89793\ 23846\ 2643 \ldots$
Definition:Decomposable Set
A set $S \subset \R^n$ is '''decomposable in $m$ sets $A_1, \ldots, A_m \subset \R^n$''' if there exist isometries $\phi_1, \ldots, \phi_m: \R^n \to \R^n$ such that: :$(1):\quad \displaystyle S = \bigcup_{k \mathop = 1}^m \phi_k \left({A_k}\right)$  :$(2):\quad \forall i \ne j: \phi_i \left({A_i}\right) \cap \phi_j \left({A_j}\right) = \varnothing$ Such a union is known as a '''decomposition'''. {{explain|The definition of Definition:Irreducible Space suggests that the definition may apply to a general topological space, not only $\R^n$. This needs to be clarified.}}
Definition:Isometry (Metric Spaces)
=== Definition 1 === {{:Definition:Isometry (Metric Spaces)/Definition 1}} === Definition 2 === {{:Definition:Isometry (Metric Spaces)/Definition 2}} Such metric spaces $M_1$ and $M_2$ are defined as being '''isometric'''. === Isometry Into === {{:Definition:Isometry (Metric Spaces)/Into}}
Definition:Equidecomposable
Two sets $S, T \subset \R^n$ are said to be '''equidecomposable''' if there exists a set: :$X = \set {A_1, \ldots, A_m} \subset \powerset {\R^n}$ where $\powerset {\R^n}$ is the power set of $\R^n$, such that both $S$ and $T$ are decomposable into the elements of $X$.
Definition:Asymptotically Equal
=== Sequences === {{:Definition:Asymptotically Equal/Sequences/Definition 1}} === Functions === {{:Definition:Asymptotically Equal/Functions}} === General Definition === {{:Definition:Asymptotically Equal/General Definition/Point}}
Definition:Inverse Sine/Arcsine
{{:Definition:Real Arcsine}}
Definition:Inverse Cosine/Real/Arccosine
{{:Graph of Arccosine Function|Graph}} From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$. From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$. Therefore, let $g: \closedint 0 \pi \to \closedint {-1} 1$ be the restriction of $\cos x$ to $\closedint 0 \pi$. Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$. This function is called '''arccosine of $x$''' and is written $\arccos x$. Thus: :The domain of $\arccos x$ is $\closedint {-1} 1$ :The image of $\arccos x$ is $\closedint 0 \pi$.
Definition:Inverse Tangent/Real/Arctangent
{{:Graph of Arctangent Function}} From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$. From the same source, we also have that: :$\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$ :$\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$ Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$. Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\R$. This function is called '''arctangent''' of $x$ and is written $\arctan x$. Thus: :The domain of $\arctan x$ is $\R$ :The image of $\arctan x$ is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Definition:Limit Point/Topology
Let $T = \left({S, \tau}\right)$ be a topological space. === Limit Point of Set === {{:Definition:Limit Point/Topology/Set}} === Limit Point of Point === The concept of a '''limit point''' can be sharpened to apply to individual points, as follows: {{:Definition:Limit Point/Topology/Point}} === Limit Point of Sequence === {{:Definition:Limit Point/Topology/Sequence}}
Definition:Vacuous Truth
Let $P \implies Q$ be a conditional statement. Suppose that $P$ is false. Then the statement $P \implies Q$ is a '''vacuous truth''', or '''is vacuously true'''. It is frequently encountered in the form: :$\forall x: \map P x \implies \map Q x$ when the propositional function $\map P x$ is false for all $x$. Such a statement is also a '''vacuous truth'''. For example, the statement: :'''All cats who are expert chess-players are also fluent in ancient Sanskrit''' is '''(vacuously) true''', because (as far as the author knows) there ''are'' no cats who are expert chess-players.
Definition:Metric Subspace
Let $\struct {A, d}$ be a metric space. Let $H \subseteq A$. Let $d_H: H \times H \to \R$ be the restriction $d \restriction_{H \times H}$ of $d$ to $H$. That is, let $\forall x, y \in H: \map {d_H} {x, y} = \map d {x, y}$. Then $d_H$ is the '''metric induced on $H$ by $d$''' or the '''subspace metric of $d$ (with respect to $H$)'''. The metric space $\struct {H, d_H}$ is called a '''metric subspace of $\struct {A, d}$'''.
Definition:Equivalent Metrics
Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$. That is, $\struct {X, d_1}$ and $\struct {X, d_2}$ are two different metric spaces on the same set $X$. Let $\sequence {x_n}$ be a sequence in $X$. Let $n \to \infty$. Suppose that $x_n \to x$ in $\struct {X, d_1}$  $x_n \to x$ in $\struct {X, d_2}$. Then $d_1$ and $d_2$ are '''equivalent metrics'''.
Definition:Taxicab Metric
The '''taxicab metric''' on $A_{1'} \times A_{2'}$ is defined as: : $d_1 \left({x, y}\right) := d_{1'} \left({x_1, y_1}\right) + d_{2'} \left({x_2, y_2}\right)$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in A_{1'} \times A_{2'}$.
Definition:Topologically Equivalent Metrics
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. === Definition 1 === {{:Definition:Topologically Equivalent Metrics/Definition 1}} === Definition 2 === {{:Definition:Topologically Equivalent Metrics/Definition 2}}
Definition:Lipschitz Equivalence
=== Metric Spaces === {{:Definition:Lipschitz Equivalence/Metric Spaces}} === Metrics === {{:Definition:Lipschitz Equivalence/Metrics/Definition 1}}
Definition:Inverse Cotangent/Real/Arccotangent
{{:Graph of Arccotangent Function}} From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$. From the same source, we also have that: :$\cot x \to + \infty$ as $x \to 0^+$ :$\cot x \to - \infty$ as $x \to \pi^-$ Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$. Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$. This function is called '''arccotangent''' of $x$ and is written $\arccot x$. Thus: :The domain of $\arccot x$ is $\R$ :The image of $\arccot x$ is $\openint 0 \pi$.
Definition:Natural Logarithm
=== Positive Real Numbers === {{:Definition:Natural Logarithm/Positive Real}} === Complex Numbers === {{:Definition:Natural Logarithm/Complex}}
Definition:Discrete Topology
Let $S \ne \O$ be a set. Let $\tau = \powerset S$ be the power set of $S$. That is, let $\tau$ be the set of all subsets of $S$: :$\tau := \set {H: H \subseteq S}$ Then $\tau$ is called '''the discrete topology on $S$''' and $\struct {S, \tau} = \struct {S, \powerset S}$ '''the discrete space on $S$''', or just '''a discrete space'''. === Finite Discrete Topology === {{:Definition:Discrete Topology/Finite}} === Infinite Discrete Topology === {{:Definition:Discrete Topology/Infinite}}
Definition:Indiscrete Topology
Let $S \ne \O$ be a set. Let $\tau = \set {S, \O}$. Then $\tau$ is called the '''indiscrete topology''' on $S$.
Definition:Fourier Transform
The '''Fourier transform''' of a Lebesgue integrable function $f: \R^N \to \C$ is the function $\map \FF f: \R^N \to \C$ given by: :$\displaystyle \map \FF {\map f \xi} := \int_{\R^N} \map f {\mathbf x} e^{-2 \pi i \mathbf x \cdot \xi} \rd \mathbf x$ for $\xi \in \R^N$. Here, the product $\mathbf x \cdot \xi$ in the exponential is the dot product of the vectors $\mathbf x$ and $\mathbf \xi$.
Definition:Metrizable Topology
Let $\struct {S, d}$ be a metric space. Let $\struct {S, \tau}$ be the topological space induced by $d$. Then for any topological space which is homeomorphic to such a $\struct {S, \tau}$, it and its topology are defined as '''metrizable'''.
Definition:Finer Topology
Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$. === Definition 1 === {{Definition:Finer Topology/Definition 1}} === Definition 2 === {{Definition:Finer Topology/Definition 2}} This can be expressed as: :$\tau_1 \ge \tau_2 := \tau_1 \supseteq \tau_2$ === Strictly Finer === {{:Definition:Finer Topology/Strictly Finer}}
Definition:Coarser Topology
Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$. Let $\tau_1 \subseteq \tau_2$. Then $\tau_1$ is said to be '''coarser''' than $\tau_2$. This can be expressed as: :$\tau_1 \le \tau_2 := \tau_1 \subseteq \tau_2$ === Strictly Coarser === {{:Definition:Coarser Topology/Strictly Coarser}}
Definition:Convolution (Measure Theory)
Let $\BB^n$ be the Borel $\sigma$-algebra on $\R^n$, and let $\lambda^n$ be Lebesgue measure on $\R^n$. === Convolution of Measurable Functions === {{:Definition:Convolution of Measurable Functions}} === Convolution of Measurable Function and Measure === {{:Definition:Convolution of Measurable Function and Measure}} === Convolution of Measures === {{:Definition:Convolution of Measures}}
Definition:Meromorphic Function
Let $U \subset \C$ be an open set. A '''meromorphic function on $U$''' is a holomorphic function on the complement of a discrete subset $P \subset U$ such that: :for all $p\in P$, $f$ has either a pole or removable singularity at $p$.
Definition:Uniform Convergence
=== Metric Space === {{:Definition:Uniform Convergence/Metric Space}} === Real Numbers === The above definition can be applied directly to the real numbers treated as a metric space: {{:Definition:Uniform Convergence/Real Numbers}} === Infinite Series === {{:Definition:Uniform Convergence/Infinite Series}}
Definition:Pointwise Convergence
Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$. Suppose that: :$\displaystyle \forall x \in D: \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$ That is: :$\forall x \in D: \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n > N: \size {\map {f_n} x - \map f x} < \epsilon$ Then '''$\sequence {f_n}$ converges to $f$ pointwise on $D$ as $n \to \infty$'''. (See the definition of convergence of a sequence).
Definition:Sub-Basis
=== Analytic Sub-Basis === {{:Definition:Sub-Basis/Analytic Sub-Basis}} === Synthetic Sub-Basis === {{:Definition:Sub-Basis/Synthetic Sub-Basis}}
Definition:Entire Function
A holomorphic self-map $f$ of the complex plane is called an '''entire function'''.
Definition:Topological Subspace
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a non-empty subset of $S$. Define: :$\tau_H := \set {U \cap H: U \in \tau} \subseteq \powerset H$ where $\powerset H$ denotes the power set of $H$. Then the topological space $T_H = \struct {H, \tau_H}$ is called a '''(topological) subspace''' of $T$.
Definition:Closure (Topology)
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition 1 === {{:Definition:Closure (Topology)/Definition 1}} === Definition 2 === {{:Definition:Closure (Topology)/Definition 2}} === Definition 3 === {{:Definition:Closure (Topology)/Definition 3}} === Definition 4 === {{:Definition:Closure (Topology)/Definition 4}} === Definition 5 === {{:Definition:Closure (Topology)/Definition 5}} === Definition 6 === {{:Definition:Closure (Topology)/Definition 6}}
Definition:Sigma-Algebra
{{:Definition:Sigma-Algebra/Definition 1}}