nyuuzyou/wikis huggingface dataset:
Waiting Time for Poisson Process obeys Exponential Distribution/Examples/Arbitrary Example 1

== Example of Use of Waiting Time for Poisson Process obeys Exponential Distribution ==
Let $P$ be a Poisson process in which events occur at a rate of $4$ per hour.
Then the mean waiting time of $P$ is $15$ minutes.
== Proof ==
From Waiting Time for Poisson Process obeys Exponential Distribution, the waiting time of $P$ is $\dfrac 1 4$.
Hence the result.
== Sources ==
* 
* 
Category:Waiting Time for Poisson Process obeys Exponential Distribution
Wald's Equation

== Theorem ==
Let $\sequence {X_n}_{n \ge 0}$ be a sequence of i.i.d. real-valued random variables.
Let $\sequence {\FF_n}_{n \ge 0}$ be the filtration generated by $\sequence {X_i}_{n \ge 0}$, that is:
:$\FF_n := \map \sigma {X_0, \ldots, X_n}$
the $\sigma$-algebra generated by $X_0, \ldots ,X_n$.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
For $n \ge 0$, let:
:$S_n := X_0 + \cdots + X_n$ 
Suppose $\expect {\size {X_0} } < \infty$.
Suppose $\expect T < \infty$.
Then:
:$\expect {S_T} = \expect {X_0} \expect T$
== Proof ==
== Sources ==
* : $2.6$: Renewal Theory
Category:Stopping Times
Wallace-Bolyai-Gerwien Theorem

== Theorem ==
Let $A$ and $B$ be plane rectilinear figures.
Then:
:there exists a dissection of $A$ into components which can be reassembled to form $B$
:
:$A$ and $B$ have the same area.
== Proof ==
== Also known as ==
== Historical Note ==
== Sources ==
* 
Category:Wallace-Bolyai-Gerwien Theorem
Category:Dissections
Category:Plane Figures
Category:Rectilineal Figures
Category:Plane Geometry
Wallace-Bolyai-Gerwien Theorem/Historical Note

== Historical Note on Wallace-Bolyai-Gerwien Theorem ==
 first formulated the theorem which would later be called the Wallace-Bolyai-Gerwien Theorem in $1807$.
 and independently proved the same thing in $1833$ and $1835$ respectively.
== Sources ==
* 
* 
* 
Category:Historical Notes
Wallis's Product

== Theorem ==
This is called Wallis's product.
== Proof 1 ==
== Wallis's Original Proof ==
== Proof 3 ==
== Also presented as ==
== Historical Note ==
== Sources ==
* : $\S 38$: Infinite Products: $38.9$: Wallis' Product
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* 
* : $\S 1.2.5$: Permutations and Factorials: Exercise $18$
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
* 
* 
* 
Category:Wallis's Product
Category:Examples of Infinite Products
Category:Formulas for Pi
Wallis's Product/Also presented as

== Wallis's Product: Also presented as == 
Wallis's Product can also be seen presented as:
:$\ds \prod_{n \mathop = 1}^\infty \frac n {n - \frac 1 2} \cdot \frac n {n + \frac 1 2} = \frac \pi 2$
Some sources present it as:
:$\ds \prod_{n \mathop = 1}^\infty \frac {4 n^2} {4 n^2 - 1} = \frac \pi 2$
which follows from the given form by an application of Difference of Two Squares on the denominator of the product.
Category:Wallis's Product
Wallis's Product/Historical Note

== Historical Note on Wallis's Product ==
Wallis's product was published by in $1656$, in his .
, of course, had no recourse to techniques.
He did this job by comparing $\ds \int_0^\pi \sin^n x \rd x$ for even and odd values of $n$, and noting that for large $n$, increasing $n$ by $1$ makes little change.
== Sources ==
* : Chapter $\text {B}.12$: Wallis's Product
* 
Category:Historical Notes
Wallis's Product/Original Proof

== Theorem ==
== Proof ==
From the Reduction Formula for Integral of Power of Sine, we have:
:$\ds (1): \quad \int \sin^n x \rd x = - \frac 1 n \sin^{n - 1} x \cos x + \frac {n - 1} n \int \sin^{n - 2} x \rd x$
Let $I_n$ be defined as:
:$\ds I_n = \int_0^{\pi / 2} \sin^n x \rd x$
As $\cos \dfrac \pi 2 = 0$ from Shape of Cosine Function, we have from $(1)$ that:
:$(2): \quad I_n = \dfrac {n-1} n I_{n - 2}$
To start the ball rolling, we note that:
:$\ds I_0 = \int_0^{\pi / 2} \rd x = \frac \pi 2$
:$\ds I_1 = \int_0^{\pi / 2} \sin x \rd x = \bigintlimits {-\cos x} 0 {\pi / 2} = 1$
We need to separate the cases where the subscripts are even and odd:
 
 
By Shape of Sine Function, we have that on $0 \le x \le \dfrac \pi 2$:
:$0 \le \sin x \le 1$
Therefore:
:$0 \le \sin^{2 n + 2} x \le \sin^{2 n +1} x \le \sin^{2 n} x$
It follows from Relative Sizes of Definite Integrals that:
:$\ds 0 < \int_0^{\pi / 2} \sin^{2 n + 2} x \rd x \le \int_0^{\pi / 2} \sin^{2 n + 1} x \rd x \le \int_0^{\pi / 2} \sin^{2 n} x \rd x$
That is:
:$(3): \quad 0 < I_{2 n + 2} \le I_{2 n + 1} \le I_{2 n}$
By $(2)$ we have:
:$\dfrac {I_{2 n + 2} } {I_{2 n} } = \dfrac {2 n + 1} {2 n + 2}$
Dividing $(3)$ through by $I_{2n}$ then, we have:
:$\dfrac {2 n + 1} {2 n + 2} \le \dfrac {I_{2 n + 1}} {I_{2 n}} \le 1$
By Squeeze Theorem, it follows that:
:$\dfrac {I_{2 n + 1} } {I_{2 n} } \to 1$ as $n \to \infty$
which is equivalent to:
:$\dfrac {I_{2 n} } {I_{2 n + 1} } \to 1$ as $n \to \infty$
Now we take $(B)$ and divide it by $(A)$ to get:
:$\dfrac {I_{2 n + 1} } {I_{2 n} } = \dfrac 2 1 \cdot \dfrac 2 3 \cdot \dfrac 4 3 \cdot \dfrac 4 5 \cdots \dfrac {2 n} {2 n - 1} \cdot \dfrac {2 n} {2 n + 1} \cdot \dfrac 2 \pi$
So:
:$\dfrac \pi 2 = \dfrac 2 1 \cdot \dfrac 2 3 \cdot \dfrac 4 3 \cdot \dfrac 4 5 \cdots \dfrac {2 n} {2 n - 1} \cdot \dfrac {2 n} {2 n + 1} \cdot \paren {\dfrac {I_{2 n} } {I_{2 n + 1} } }$
Taking the limit as $n \to \infty$ gives the result.
== Sources ==
* : Chapter $\text {B}.12$: Wallis's Product
Category:Wallis's Product
Wallis's Product/Proof 1

== Theorem ==
== Proof ==
Into Euler Formula for Sine Function:
we substitute $x = \dfrac \pi 2$.
From Sine of Half-Integer Multiple of Pi:
:$\sin \dfrac \pi 2 = 1$
Hence:
 
== Sources ==
* : Chapter $\text {A}.21$: Euler ()
* : Chapter $\text {B}.12$: Wallis's Product
* : Chapter $\text {B}.14$: Euler's Discovery of the Formula $\ds \sum_1^\infty \frac 1 {n^2} = \frac {\pi^2} 6$
Category:Wallis's Product
Wang's Paradox

== Theorem ==
$0$ is a small number.
If $n$ is a small number, then $n + 1$ is a small number.
By the Principle of Mathematical Induction, it follows that all numbers are small numbers.
== Resolution ==
There is no paradox.
Compared to a number which is much greater, every number is a small number.
Besides, small is an imprecise predicate, undefined in mathematics, about which no solid logical conclusions can be inferred.
== Historical Note ==
== Sources ==
* 
Category:Veridical Paradoxes
Category:Sorites
Water and Wine Problem

== Problem ==
You are presented with two glasses of similar size.
One glass contains . 
The other contains water in like volume to the wine.
One spoonful of wine is taken from the wine glass and mixed into the water.
Then an identically-sized spoonful of the water-wine mixture is transferred from the water glass into the wine glass.
Which mixture is purer: the one in the water glass, or the one in the wine glass?
== Solution ==
Let $x$ be the volume of liquid held by a glass.
Let $y$ be the volume of liquid held by the spoon.
After the first operation during which a spoonful of wine is transferred to the water glass:
: the volume of wine in the wine glass will be $x - y$
: the volume of liquid in the water glass will be $x + y$, of which $x$ is water.
During the course of the second operation, where a spoonful of wine-water mixture is transferred back to the wine glass, let $z$ be the volume of wine in the spoon.
The volume of water in the spoon is therefore $y - z$.
Thus, while $z$ is the volume of wine being added to the wine glass, $y - z$ is the volume of water being removed from the water glass.
After the second operation:
: the volume of wine in the wine glass is $\left({x - y}\right) + z$
: the volume of water in the water glass is $x - \left({y - z}\right) = \left({x - y}\right) + z$.
The mixtures are therefore of equal purity.
Category:Classic Problems
Watson's Hypergeometric Theorem

== Theorem ==
Let $a, b, c \in \C$.
Then:
:$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {\dfrac 1 2 \paren {a + b + 1}, 2 c } } \, \middle \vert \, 1} = \dfrac {\map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 1 2 + c} \map \Gamma {\dfrac 1 2 \paren {1 + a + b} } \map \Gamma {\dfrac 1 2 \paren {1 - a - b} + c } } {\map \Gamma {\dfrac 1 2 \paren {1 + a} } \map \Gamma {\dfrac 1 2 \paren {1 + b} } \map \Gamma {\dfrac 1 2 \paren {1 - a} + c } \map \Gamma {\dfrac 1 2 \paren {1 - b} + c } }$
where:
:${}_3 \operatorname F_2$ is the generalized hypergeometric function
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
== Proof ==
From Thomae's Transformation, we have:
The generalized hypergeometric function on the can be summed using Dixon's Hypergeometric Theorem.
From Dixon's Hypergeometric Theorem, we have:
:$\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} } $
Let:
Then:
Making these substitutions to the generalized hypergeometric function on the , we have:
We now have:
From Legendre's Duplication Formula, we have:
Therefore:
== Also known as ==
Some sources refer to Watson's Hypergeometric Theorem as Watson's Theorem.
== Also see ==
* Dixon's Hypergeometric Theorem
* Gauss's Hypergeometric Theorem
* Kummer's Hypergeometric Theorem
* Properties of Generalized Hypergeometric Function
== Sources ==
* Chapter $\text {3}$. The hypergeometric series
* 
Category:Watson's Hypergeometric Theorem
Category:Hypergeometric Functions
Wave Equation/Examples/Harmonic Wave

== Examples of Use of the Wave Equation ==
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$ and without change of shape.
From Equation of Harmonic Wave, the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$(1): \quad \map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }$
where:
:$x$ denotes the displacement from the origin along the $x$-axis
:$t$ denotes the time
:$\lambda$ is the wavelength of $\phi$
:$\tau$ is the period of $\phi$.
This equation satisfies the wave equation.
== Proof ==
The wave equation is expressible as:
:$\dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2} = \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}$
We have by partial differentiation:
and:
As $y$ and $z$ do not appear in $(1)$, the partial derivative of $(1)$ $y$ and $z$ is identically zero.
Hence we have:
:$\dfrac {\partial^2 \phi} {\partial t^2} = c^2 \paren {\dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2} }$
and the result follows.
</onlyinclude>
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 5$
Category:Examples of Use of Wave Equation
Wave Equation/Examples/Wave with Constant Velocity

== Examples of Use of the Wave Equation ==
Let $\phi$ be a wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$ and without change of shape.
From Equation of Wave with Constant Velocity, the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$(1): \quad \map \phi {x, t} = \map f {x - c t}$
where:
:$x$ denotes the displacement from the origin along the $x$-axis
:$t$ denotes the time.
This equation satisfies the wave equation.
== Proof ==
The wave equation is expressible as:
:$\dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2} = \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}$
We have by partial differentiation of $(1)$:
and:
As $y$ and $z$ do not appear in $(1)$, the partial derivative of $(1)$ $y$ and $z$ is identically zero.
Hence we have:
:$\dfrac {\partial^2 \phi} {\partial t^2} = c^2 \paren {\dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2} }$
and the result follows.
</onlyinclude>
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 5$
Category:Examples of Use of Wave Equation
Wave Equation is Linear

== Theorem ==
Let $\phi_1$ and $\phi_2$ be particular solutions to the wave equation.
Then:
:$a_1 \phi_1 + a_2 \phi_2$ is also a particular solution to the wave equation.
== Proof ==
Hence:
and it is seen that $a_1 \phi_1 + a_2 \phi_2$ also satisfies the wave equation.
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 6$
Category:Wave Equation
Wave Number of Harmonic Wave

== Theorem ==
Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The wave number $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the wavelength of $\phi$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(10)$
Category:Wave Number of Harmonic Wave
Category:Wave Number
Category:Harmonic Waves
Wave Number of Harmonic Wave/Proof 2

== Theorem ==
== Proof ==
By definition, $k$ is the number of complete wavelengths of $\phi$ per unit displacement along the $x$-axis.
By definition, $\lambda$ is the period of the wave profile of $\phi$.
So between two points at unit distance apart, there are $\dfrac 1 \lambda$ wavelengths of $\phi$.
The result follows by definition of wave number.
Category:Wave Number of Harmonic Wave
Wave Number of Periodic Wave

== Theorem ==
Let $\phi$ be a periodic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
The wave number $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the wavelength of $\phi$.
== Proof ==
By definition, $k$ is the number of complete wavelengths of $\phi$ per unit displacement along the $x$-axis.
By definition, $\lambda$ is the period of the wave profile of $\phi$.
So between two points at unit distance apart, there are $\dfrac 1 \lambda$ wavelengths of $\phi$.
The result follows by definition of wave number.
== Examples ==
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$
* 
* 
Category:Wave Number of Periodic Wave
Category:Wave Number
Category:Periodic Waves
Wave Profile of Harmonic Wave

== Theorem ==
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the wave profile of $\phi$ can be expressed as:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
== Proof ==
By definition, a harmonic wave is a wave whose wave profile can be expressed as a sine curve.
By definition, a sine curve can be expressed in the form:
:$\map \phi x = a \map \sin {\omega x + \epsilon}$
where $a$, $\omega$ and $\epsilon$ are arbitrary constants.
We select $\epsilon$ so as to set:
:$\epsilon = \dfrac \pi 2$
We then have:
Hence at time $t = 0$ the equation expressing a wave in the form of a sine curve is:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
Hence the result by definition of wave profile.
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$
Category:Harmonic Waves
Category:Wave Profiles
Wavelength of Harmonic Wave

== Theorem ==
Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The wavelength $\lambda$ of $\phi$ can be expressed as:
:$\lambda = \dfrac {2 \pi} \omega$
== Proof ==
By definition, $\lambda$ is the period of the wave profile of $\phi$.
From Wave Profile of Harmonic Wave, the wave profile of $\phi$ is given by:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
From Period of Real Cosine Function:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos {\omega x + 2 \pi}$
So the period of $\paren {\map \phi x}_{t \mathop = 0}$ is $\dfrac {2 \pi} \omega$.
Hence the result by definition of wavelength.
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$
Category:Wavelength
Category:Harmonic Waves
Way Above Closure is Open

== Theorem ==
Let $L = \struct {S, \preceq, \tau}$ be a complete continuous topological lattice with Scott topology.
Let $x \in S$.
Then $x^\gg$ is open
where $x^\gg$ denotes the way above closure of $x$.
== Proof ==
By Way Above Closure is Upper:
:$x^\gg$ is upper.
We will prove that
:$x^\gg$ is inaccessible by directed suprema.
Let $D$ be a directed subset of $S$ such that
:$\sup D \in x^\gg$
By definition of way above closure:
:$x \ll \sup D$
By Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element:
:$\exists d \in D: x \ll d$
By definition of way above closure:
:$d \in x^\gg$
By definitions of intersection and non-empty set:
:$x^\gg \cap D \ne \O$
Thus by definition of Scott topology:
:$x^\gg$ is open.
== Sources ==
* 
* 
Category:Topological Order Theory
Category:Continuous Lattices
Way Above Closure is Subset of Upper Closure of Element

== Theorem ==
Let $\struct {S, \preceq}$ be an ordered set.
Let $x \in S$.
Then:
:$x^\gg \subseteq x^\succeq$
where
:$x^\gg$ denotes the way above closure of $x$
:$x^\succeq$ denotes the upper closure of $x$.
== Proof ==
Let $y \in x^\gg$.
By definition of way above closure:
:$x \ll y$
where $\ll$ denotes the way below relation.
By Way Below implies Preceding:
:$x \preceq y$
Thus by definition of upper closure of element:
:$y \in x^\succeq$
== Sources ==
* 
* 
Category:Upper Closures
Way Above Closure is Upper

== Theorem ==
Let $\struct {S, \preceq}$ be an ordered set.
Let $x \in S$.
Then $x^\gg$ is upper
where $x^\gg$ denotes the way above closure of $x$.
== Proof ==
Let $y \in x^\gg$, $z \in S$ such that
:$y \preceq z$
By definition of way above closure:
:$x \ll y$
By Preceding and Way Below implies Way Below:
:$x \ll z$
Thus by definition of way above closure:
:$z \in x^\gg$
== Sources ==
* 
Category:Way Below Relation
Way Above Closures Form Basis

== Theorem ==
Let $L = \struct {S, \preceq, \tau}$ be a complete continuous topological lattice with Scott topology.
Then $\set {x^\gg: x \in S}$ is an (analytic) basis of $L$.
== Proof ==
Define $B = \set {x^\gg: x \in S}$.
Thus by Way Above Closure is Open:
:$B \subseteq \tau$
We will prove that:
:for all $x \in S$: there exists a local basis $Q$ of $x$: $Q \subseteq B$
Let $x \in S$.
By Way Above Closures that Way Below Form Local Basis:
:$Q := \set {g^\gg: g \in S \land g \ll x}$ is a local basis at $x$.
Thus by definition of subset:
:$Q \subseteq B$
Thus by Characterization of Analytic Basis by Local Bases:
:$B$ is an (analytic) basis of $L$.
== Sources ==
* 
* 
Category:Topological Order Theory
Category:Continuous Lattices
Way Above Closures that Way Below Form Local Basis

== Theorem ==
Let $L = \struct {S, \preceq, \tau}$ be a complete continuous topological lattice with Scott topology.
Let $p \in S$.
Then $\set {q^\gg: q \in S \land q \ll p}$ is a local basis at $p$.
== Proof ==
Define $B := \set {q^\gg: q \in S \land q \ll p}$
By Way Above Closure is Open:
:$B \subseteq \tau$
By definition of way above closure:
:$\forall X \in B: p \in X$
Thus by definition:
:$B$ is set of open neighborhoods.
Let $U$ be an open subset of $S$ such that
:$p \in U$
By Open implies There Exists Way Below Element:
:$\exists u \in U: u \ll p$
Thus by definition of $B$:
:$u^\gg \in B$
By definition of Scott topology:
:$U$ is upper.
We will prove that
:$u^\gg \subseteq U$
Let $z \in u^\gg$
By definition of way above closure:
:$u \ll z$
By Way Below implies Preceding:
:$u \preceq z$
Thus by definition of upper section:
:$z \in U$
== Sources ==
* 
* 
Category:Topological Order Theory
Category:Way Below Relation
Category:Continuous Lattices
Way Below Closure is Directed in Bounded Below Join Semilattice

== Theorem ==
Let $\struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $x \in S$.
Then
:$x^\ll$ is directed.
== Proof ==
By Bottom is Way Below Any Element:
:$\bot \ll x$
By definition of way below closure:
:$\bot \in x^\ll$
Thus by definition:
:$x^\ll$ is a non-empty set.
Let $y, z \in x^\ll$
By definition of way below closure:
:$y \ll x$ and $z \ll x$
By Join is Way Below if Operands are Way Below
:$y \vee z \ll x$
By definition of way below closure:
:$y \vee z \in x^\ll$
By Join Succeeds Operands:
:$y \preceq y \vee z$ and $z \preceq y \vee z$
Thus by definition
:$x^\ll$ is directed.
== Sources ==
* 
* 
* 
Category:Join and Meet Semilattices
Category:Way Below Relation
Way Below Closure is Ideal in Bounded Below Join Semilattice

== Theorem ==
Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $x \in S$.
Then
:$x^\ll$ is ideal in $L$.
== Proof ==
By Way Below Closure is Directed in Bounded Below Join Semilattice:
:$x^\ll$ is a non-empty directed set.
Let $y \in x^\ll, z \in S$ such that
:$z \preceq y$
By definition of way below closure:
:$y \ll x$
By definition of reflexivity:
:$x \preceq x$
By Preceding and Way Below implies Way Below:
:$z \ll x$
Thus by definition of way below closure:
:$z \in x^\ll$
Thus by definition
:$x^\ll$ is a lower section.
Thus by definition
:$x^\ll$ is an ideal in $L$.
== Sources ==
* 
* 
* 
* 
Category:Join and Meet Semilattices
Category:Way Below Relation
Way Below Closure is Lower Section

== Theorem ==
Let $L = \struct {S, \vee, \preceq}$ be an ordered set.
Let $x \in S$.
Then
:$x^\ll$ is a lower section.
== Proof ==
Let $y \in x^\ll, z \in S$ such that:
:$z \preceq y$
By definition of way below closure:
:$y \ll x$
By definition of reflexivity:
:$x \preceq x$
By Preceding and Way Below implies Way Below:
:$z \ll x$
Thus by definition of way below closure:
:$z \in x^\ll$
Thus by definition:
:$x^\ll$ is a lower section.
== Sources ==
* 
* 
Category:Join and Meet Semilattices
Category:Way Below Relation
Category:Lower Sections
Way Below Compact is Topological Compact

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $L = \struct {\tau, \preceq}$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$
Let $x \in \tau$.
Then
:$x$ is compact in $L$ (it means: $x \ll x$)
:$T_x$ is compact (topologically)
where $T_x = \struct {x, \tau_x}$ denotes the topological subspace of $x$.
== Proof ==
=== Sufficient Condition ===
Let
:$x \ll x$
Let $F \subseteq \tau_x$ be a open cover of $x$.
By definition of cover:
:$x \subseteq \bigcup F$
By definition of topological space:
:$\forall y \in \tau: x \cap y \in \tau$
By definition of subset:
:$\tau_x \subseteq \tau$
By Subset Relation is Transitive:
:$F \subseteq \tau$
By Way Below in Ordered Set of Topology:
:there exists a finite subset $G$ of $F$: $\ds x \subseteq \bigcup G$
By definition:
:$G$ is cover of $x$
Thus by definition:
:$x$ has finite subcover.
Thus by definition:
:$T_x$ is compact.
=== Necessary Condition ===
Let
:$T_x$ is compact.
Let $F$ be a set of open subsets of $S$ such that
:$\ds x \subseteq \bigcup F$
Define $Y := \set {x \cap y: y \in F}$
By definition of subset:
:$Y \subseteq \tau_x$
By definition of union:
:$\ds x \subseteq \bigcup Y$
By definition
:$Y$ is open cover of $x$.
By definition of compact:
:$x$ has finite subcover $G$.
By Axiom of Choice:
:$\exists g: G \to F: \forall y \in G: \map g y \cap x = y$
By Cardinality of Image of Set not greater than Cardinality of Set:
:$\card {\map {g^\to} G} \le \card G$
where $\card G$ denotes the cardinality of $G$.
Thus
:$\map {g^\to} G$ is finite.
Thus by definitions of subset and image of set:
:$\map {g^\to} G \subseteq F$
By Intersection is Subset:
:$\forall y \in G: y \subseteq \map g y$
By Set Union Preserves Subsets/Families of Sets:
:$\ds \bigcup G = \bigcup_{y \mathop \in G}y \subseteq \bigcup_{y \mathop \in G} \map g y = \bigcup \map {g^\to} G$
By definition of cover:
:$\ds x \subseteq \bigcup G$
Thus by Subset Relation is Transitive:
:$\ds x \subseteq \bigcup \map {g^\to} G$
Thus by Way Below in Ordered Set of Topology;
:$x \ll x$
== Sources ==
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Category:Way Below Relation
Category:Topology