hcju/proofwikips huggingface dataset:
Closure of Intersection and Symmetric Difference imply Closure of Union
Let $\mathcal R$ be a system of sets such that for all $A, B \in \mathcal R$: : $(1): \quad A \cap B \in \mathcal R$ : $(2): \quad A * B \in \mathcal R$ where $\cap$ denotes set intersection and $*$ denotes set symmetric difference. Then: :$\forall A, B \in \mathcal R: A \cup B \in \mathcal R$ where $\cup$ denotes set union.
Complement of Limit Inferior is Limit Superior of Complements
Let $\left\{{E_n : n \in \N}\right\}$ be a sequence of sets. Then: :$\displaystyle \complement \left({\liminf_{n \mathop \to \infty} \ E_n}\right) = \limsup_{n \mathop \to \infty} \ \complement \left({E_n}\right)$ where $\liminf$ and $\limsup$ denote the limit inferior and limit superior, respectively.
Sigma-Ring is Closed under Countable Intersections
Let $\mathcal R$ be a $\sigma$-ring. Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N} \in \mathcal R$ be a sequence of sets in $\mathcal R$. Then: :$\displaystyle \bigcap_{n \mathop = 1}^\infty A_n \in \mathcal R$
Pappus's Hexagon Theorem
Let $A, B, C$ be a set of collinear points. Let $a, b, c$ be another set of collinear points. Let $X, Y, Z$ be the points of intersection of each of the straight lines $Ab$ and $aB$, $Ac$ and $aC$, and $Bc$ and $bC$. Then $X, Y, Z$ are collinear points.
Two Planes have Line in Common
Two distinct planes have exactly one (straight) line in common.
Principle of Duality in the Plane
Let $P$ be a theorem of projective geometry proven using the propositions of incidence. Let $Q$ be the statement created from $P$ by interchanging: :$(1): \quad$ the terms '''point''' and '''(straight) line''' :$(2): \quad$ the terms '''collinear''' (of points) and '''concurrent''' (of lines) :$(3): \quad$ the terms '''lie on''' and '''intersect at''' and so on. Then $Q$ is also a theorem of projective geometry.
Principle of Duality in Space
Let $P$ be a theorem of projective geometry proven using the propositions of incidence. Let $Q$ be the statement created from $P$ by interchanging: :$(1) \quad$ the terms '''point''' and '''plane''' :$(2) \quad$ the terms '''lie on''' and '''intersect at''' and so on. Then $Q$ is also a theorem of projective geometry.
Desargues' Theorem
Let $\triangle ABC$ and $\triangle A'B'C'$ be triangles lying in the same or different planes. Let the lines $AA'$, $BB'$ and $CC'$ intersect in the point $O$. Then $BC$ meets $B'C'$ in $L$, $CA$ meets $C'A'$ in $M$ and $AB$ meets $A'B'$ in $N$, where $L, M, N$ are collinear.
Domain of Composite Mapping
Let $S_1, S_2, S_3$ be sets. Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings. Let $f_2 \circ f_1: S_1 \to S_3$ be the composite mapping of $f_1$ and $f_2$. Then: :$\Dom {f_1} = \Dom {f_2 \circ f_1}$ where $\Dom {f_1}$ denotes the domain of $f_1$.
Union of Functions Theorem
Let $X$ be a set. Let $\set {X_i: i \in \N}$ be an exhausting sequence of sets on $X$. For each $i \in \N$, let $g_i: X_i \to Y$ be a mapping such that: :$g_{i + 1} \restriction X_i = g_i$ where $g_{i + 1} \restriction X_i$ denotes the restriction of $g_{i + 1}$ to $g_i$. Then: :$\displaystyle \bigcup \set {g_i: i \in \N}$ is a mapping from $X$ to $Y$.
Inductive Definition of Sequence
Let $X$ be a set. Let $h \in \N$. Let $a_i \in X$ for all $i \in \set {1, 2, \ldots, h}$. Let $S$ be the set of all finite sequences whose codomains are in $X$. Let $G: S \to X$ be a mapping. Then there is a unique sequence $f$ whose codomain is in $X$ such that: :$f_i = \begin{cases} a_i & : i \in \set {1, 2, \ldots, h} \\ \map G {f_1, f_2, \ldots, f_{i - 1} } & : i \ge h + 1 \end{cases}$
Model of Root of Propositional Tableau is Model of Branch
Let $\left({T, \mathbf H, \Phi}\right)$ be a propositional tableau. Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a boolean interpretation such that: :$v \models_{\mathrm{BI}} \mathbf H$ that is, such that $v$ is a model for the root $\mathbf H$ of $T$. Then there exists a branch $\Gamma$ of $T$ such that: :$v \models_{\mathrm{BI}} \Phi \left[{\Gamma}\right]$ where $\Phi \left[{\Gamma}\right]$ denotes the image of $\Gamma$ under $\Phi$.
Antireflexive and Transitive Relation is Antisymmetric
Let $\mathcal R \subseteq S \times S$ be a relation which is not null. Let $\mathcal R$ be antireflexive and transitive. Then $\mathcal R$ is also antisymmetric.
Tableau Confutation implies Unsatisfiable
Let $\mathbf H$ be a collection of WFFs of propositional logic. Suppose there exists a tableau confutation of $\mathbf H$. Then $\mathbf H$ is unsatisfiable for boolean interpretations.
Finished Branch Lemma
Let $\Gamma$ be a finished branch of a propositional tableau $\left({T, \mathbf H, \Phi}\right)$. Let $v$ be a boolean interpretation such that: :$v \models_{\mathrm{BI}} \mathbf A$ for every basic WFF $\mathbf A$ that occurs along $\Gamma$. Then: :$v \models_{\mathrm{BI}} \Phi \left[{\Gamma}\right]$ where $\Phi \left[{\Gamma}\right]$ is the image of $\Gamma$ under $\Phi$.
Diagonal Relation is Serial
Let $S$ be a set. Let $\Delta_S$ be the diagonal relation on $S$. Then $\Delta_S$ is a serial relation.
Serial Relation is not Null
Let $S$ be a set such that $S \ne \varnothing$. Let $\mathcal R$ be a serial relation on $S$. Then $\mathcal R$ is not a null relation.
Transitive Relation is Antireflexive iff Asymmetric
Let $\mathcal R \subseteq S \times S$ be a relation which is not null. Let $\mathcal R$ be transitive. Then $\mathcal R$ is antireflexive iff $\mathcal R$ is asymmetric.
Symmetric and Antisymmetric Relation is Transitive
Let $S$ be a set. Let $\RR \subseteq S \times S$ be a relation in $S$ which is both symmetric and antisymmetric. Then $\RR$ is transitive.
Relation is Reflexive and Coreflexive iff Diagonal
Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a relation on $S$. Then $\mathcal R$ is reflexive and coreflexive iff: : $\mathcal R = \Delta_S$ where $\Delta_S$ is the diagonal relation.
Symmetric Preordering is Equivalence Relation
Let $\RR \subseteq S \times S$ be a preordering on a set $S$. Let $\RR$ also be symmetric. Then $\RR$ is an equivalence relation on $S$.
Antisymmetric Preordering is Ordering
Let $\mathcal R \subseteq S \times S$ be a preordering on a set $S$. Let $\mathcal R$ also be antisymmetric. Then $\mathcal R$ is an ordering on $S$.
Relation is Antisymmetric and Reflexive iff Intersection with Inverse equals Diagonal Relation
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. Then $\mathcal R$ is both antisymmetric and reflexive iff: :$\mathcal R \cap \mathcal R^{-1} = \Delta_S$ where $\Delta_S$ denotes the diagonal relation.
Reflexive and Transitive Relation is Idempotent
Let $\RR \subseteq S \times S$ be a relation on a set $S$. Let $\RR$ be both reflexive and  transitive. Then $\RR$ is idempotent, in the sense that: :$\RR \circ \RR = \RR$ where $\circ$ denotes composition of relations.
Equivalence of Definitions of Ordering/Proof 2
The following definitions of ordering are equivalent:
Ordered Tuple/Equality
Let: :$(1): \quad \sequence {a_m} = \tuple {a_1, a_2, \ldots, a_m}$ and :$(2): \quad \sequence {b_n} = \tuple {b_1, b_2, \ldots, b_n}$ be ordered tuples for some $m, n \in \N_{>0}$. Then: :$\sequence {a_m} = \sequence {b_n} \iff n = m \land \forall j \in \N^*_n: a_j = b_j$
Image of Domain of Relation is Image Set
Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation. The image of the domain of $\mathcal R$ is the image set of $\mathcal R$: :$\mathcal R \left [{\operatorname{Dom} \left({\mathcal R}\right)}\right] = \operatorname{Im} \left ({\mathcal R}\right)$ where $\operatorname{Im} \left ({\mathcal R}\right)$ is the image of $\mathcal R$.
Condition for Relation to be Transitive and Antitransitive
Let $S$ be a set. Let $\RR \subseteq S \times S$ be a relation in $S$. Then: : $\RR$ is both transitive and antitransitive : : $\neg \paren {\exists x, y, z \in S: x \mathrel {\RR} y \land y \mathrel {\RR} z}$
Transitive and Antitransitive Relation is Asymmetric
Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a relation in $S$. Let $\mathcal R$ be both transitive and antitransitive. Then $\mathcal R$ is asymmetric.
Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise
Let $Q$ be a valid categorical syllogism in Figure $\text I$. Then it is a necessary condition that: :The major premise of $Q$ be a universal categorical statement and :The minor premise of $Q$ be an affirmative categorical statement.
Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise
Let $Q$ be a valid categorical syllogism in Figure $\text{II}$. Then it is a necessary condition that: :The major premise of $Q$ be a universal categorical statement and :The conclusion of $Q$ be a negative categorical statement.
Valid Syllogism in Figure III needs Particular Conclusion and if Negative then Negative Major Premise
Let $Q$ be a valid categorical syllogism in Figure $\text {III}$. Then it is a necessary condition that: :The conclusion of $Q$ be a particular categorical statement and: :If the conclusion of $Q$ be a negative categorical statement, then so is the major premise of $Q$.
Valid Syllogisms in Figure IV
Let $Q$ be a valid categorical syllogism in Figure $\text {IV}$. Then it is a necessary condition that: :$(1): \quad$ Either: :: the major premise of $Q$ be a negative categorical statement :or: :: the minor premise of $Q$ be a universal categorical statement :or both. :$(2): \quad$ If the conclusion of $Q$ be a negative categorical statement, then the major premise of $Q$ be a universal categorical statement. :$(3): \quad$ If the conclusion of $Q$ be a universal categorical statement, then the minor premise of $Q$ be a negative categorical statement.
Elimination of all but 24 Categorical Syllogisms as Invalid
Of the $256$ different types of categorical syllogism, all but $24$ can be identified as invalid. These are the $24$ patterns which may still be valid: :$\begin{array}{rl} \text{I} & AAA \\ \text{I} & AII \\ \text{I} & EAE \\ \text{I} & EIO \\ \text{I} & AAI \\ \text{I} & EAO \\ \end{array} \qquad \begin{array}{rl} \text{II} & EAE \\ \text{II} & AEE \\ \text{II} & AOO \\ \text{II} & EIO \\ \text{II} & EAO \\ \text{II} & AEO \\ \end{array} \qquad \begin{array}{rl} \text{III} & AAI \\ \text{III} & AII \\ \text{III} & IAI \\ \text{III} & EAO \\ \text{III} & EIO \\ \text{III} & OAO \\ \end{array} \qquad \begin{array}{rl} \text{IV} & AAI \\ \text{IV} & AEE \\ \text{IV} & EAO \\ \text{IV} & EIO \\ \text{IV} & IAI \\ \text{IV} & AEO \\ \end{array}$
Extension of Contradictory Branch is Contradictory
Let $T$ be a propositional tableau. Let $\Gamma$ be a contradictory branch of $T$. Let $\Gamma'$ be an extension of $\Gamma$. Then $\Gamma'$ is also contradictory.
Finished Propositional Tableau has Finished Branch or is Confutation
Let $\struct {T, \mathbf H, \Phi}$ be a finished propositional tableau. Then one of the following holds: :$T$ has a finished branch :$T$ is a tableau confutation.
Definition:P-Series
Let $p \in \C$ be a complex number. The series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^p} = 1 + \frac 1 {2^p} + \frac 1 {3^p} + \frac 1 {4^p} + \dotsb$ is known as a '''$p$-series'''.
Alternating Harmonic Series is Conditionally Convergent
The alternating harmonic series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \cdots$ is conditionally convergent.
Manipulation of Absolutely Convergent Series/Permutation
If $\pi: \N \to \N$ is a permutation of $N$, then: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty a_{\map \pi n}$
Manipulation of Absolutely Convergent Series/Characteristic Function
Let $A \subseteq \N$. Then: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n \map {\chi_A} n = \sum_{n \mathop \in A} a_n$ where $\chi_A$ is the characteristic function of $A$.
Manipulation of Absolutely Convergent Series/Scale Factor
Let $c \in \R$, or $c \in \C$. Then: :$\displaystyle c \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty c a_n$
Cauchy Product of Absolutely Convergent Series
Let $\displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 0}^\infty b_n$ be two real series that are absolutely convergent. Then the Cauchy product of $\displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 0}^\infty b_n$ is absolutely convergent.
Binomial Coefficient with Self minus One
:$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$
Rising Sum of Binomial Coefficients/Marginal Cases
=== $n = 0$ === When $n = 0$ we have: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^m \binom j 0       | r = \binom 0 0 + \binom 1 0 + \binom 2 0 + \cdots \binom m 0       | c =  }} {{eqn | r = 1 + 1 + \cdots + 1       | c = from $0$ to $m$ }} {{eqn | r = m + 1       | c = as there are $m+1$ of them }} {{eqn | r = \binom {m+1} 1       | c =  }} {{end-eqn}} So the theorem holds for $n = 0$.  === $n = 1$ === When $n = 1$ we have: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^m \binom {1 + j} 1       | r = \binom 1 1 + \binom 2 1 + \binom 3 1 + \cdots \binom {m + 1} 1       | c =  }} {{eqn | r = 1 + 2 + \cdots + \paren {m + 1}       | c =  }} {{eqn | r = \frac {\paren {m + 1} \paren {m + 2} } 2       | c = Closed Form for Triangular Numbers }} {{eqn | r = \binom {m + 2} 2       | c = {{Defof|Binomial Coefficient}} }} {{end-eqn}} So the theorem holds for $n = 1$.
Uniformly Convergent Series of Continuous Functions is Continuous
Let $\sequence {f_n}$ be a sequence of real functions. Let each of $\sequence {f_n}$ be continuous on the interval $\hointr a b$. {{explain|Investigation needed as to whether there is a mistake in {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition}} -- should it actually be a closed interval?}} Let the series: :$\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$ be uniformly convergent for all $x \in \closedint a b$. Then $f$ is continuous on $\hointr a b$.
Definite Integral of Uniformly Convergent Series of Continuous Functions
Let $\sequence {f_n}$ be a sequence of real functions. Let each of $\sequence {f_n}$ be continuous on the interval $\hointr a b$. {{explain|Investigation needed as to whether there is a mistake in {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition}} -- should it actually be a closed interval?}} Let the series: :$\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$ be uniformly convergent for all $x \in \closedint a b$. Then: :$\displaystyle \int_a^b \map f x \rd x = \sum_{n \mathop = 1}^\infty \int_a^b \map {f_n} x \rd x$
Derivative of Uniformly Convergent Series of Continuously Differentiable Functions
Let $\sequence {f_n}$ be a sequence of real functions. Let each of $\sequence {f_n}$ be continuously differentiable on the interval $\closedint a b$. Let the series: :$\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$ be pointwise convergent for all $x \in \closedint a b$. Let the series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac \d {\d x} \map {f_n} x$ be uniformly convergent for all $x \in \closedint a b$. Then: :$\displaystyle \frac \d {\d x} \map f x := \sum_{n \mathop = 1}^\infty \frac \d {\d x} \map {f_n} x$
Power Series Converges Uniformly within Radius of Convergence
Let $\displaystyle S := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Let $\rho \in \R$ such that $0 \le \rho < R$. Then $S$ is uniformly convergent on $D = \set {x: \size {x - \xi} \le \rho}$.
Power Series Converges to Continuous Function
Let $\displaystyle \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Then $\map f x$ is a continuous function on $\set {x: \size {x - \xi} < R}$.
Power Series is Termwise Integrable within Radius of Convergence
Let $\displaystyle \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Then: :$\displaystyle \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x = \sum_{n \mathop = 0}^\infty a_n \frac {x^{n + 1} } {n + 1}$
Power Series is Termwise Differentiable within Radius of Convergence
Let $\displaystyle \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Then: :$\displaystyle \frac \d {\d x} \map f x = \sum_{n \mathop = 0}^\infty \frac \d {\d x} a_n x^n = \sum_{n \mathop = 1}^\infty  n a_n x^{n - 1}$
Power Series Expansion for Logarithm of 1 + x
The Newton-Mercator series defines the  natural logarithm function as a power series expansion: {{begin-eqn}} {{eqn | l = \map \ln {1 + x}       | r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n }} {{eqn | r = x - \frac {x^2} 2 + \frac {x^3} 3 - \frac {x^4} 4 + \cdots }} {{end-eqn}} valid for all $x \in \R$ such that $-1 < x \le 1$.
Newton-Mercator Series/Examples/2
The Newton-Mercator Series for $x = 1$ converges to the natural logarithm of $2$: {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n       | r = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb       | c =  }} {{eqn | r = \ln 2       | c =  }} {{end-eqn}} This real number is known as Mercator's constant.
Power Series Expansion for Sine Function
The sine function has the power series expansion: {{begin-eqn}} {{eqn | l = \sin x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.
Power Series Expansion for Cosine Function
The cosine function has the power series expansion: {{begin-eqn}} {{eqn | l = \cos x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.
Spacing Limit Theorem
Let $X_{\left({i}\right)}$ be the $i$th ordered statistic of $N$ samples from a continuous random distribution with density function $f_X \left({x}\right)$. Then the spacing between the ordered statistics given $X_{\left({i}\right)}$ converges in distribution to exponential for sufficiently large sampling according to: :$N \left({X_{\left({i + 1}\right)} - X_{\left({i}\right)} }\right) \xrightarrow D \exp \left({ \dfrac 1 { f \left({ X_{\left({i}\right)} }\right)} }\right)$ as $N \to \infty$ for $i = 1, 2, 3, \dotsc, N-1$.
Union of Closures of Singleton Rationals is Rational Space
Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$. Let $B_\alpha$ denote the singleton containing the rational number $\alpha$. Then the union of the closures in the set of real numbers $\R$ of all $B_\alpha$ is $\Q$: :$\displaystyle \bigcup_{\alpha \mathop \in \Q} \map \cl {B_\alpha} = \Q$
Closure of Union of Singleton Rationals is Real Number Line
Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$. Let $B_\alpha$ be the singleton containing the rational number $\alpha$. Then the closure in the set of real numbers $\R$ of the union of all $B_\alpha$ is $\R$ itself: :$\displaystyle \map \cl {\bigcup_{\alpha \mathop \in \Q} B_\alpha} = \R$
Closure of Rational Numbers is Real Numbers
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology. Then: :$\Q^- = \R$ where $\Q^-$ denotes the closure of $\Q$.
Real Number is Closed in Real Number Line
Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Let $\alpha \in \R$ be a real number. Then $\set \alpha$ is closed in $\struct {\R, \tau}$.
Intersection of Exteriors of Singleton Rationals is Irrationals
Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$. Let $B_\alpha$ be the singleton containing the rational number $\alpha$. Then: :$\displaystyle \bigcap_{\alpha \mathop \in \Q} B_\alpha^e = \R \setminus \Q$ where $B_\alpha^e$ denotes the exterior of $B_\alpha$ in $\R$.
Rational Numbers form F-Sigma Set in Reals
Let $\Q$ be the set of rational numbers. Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Then $\Q$ is a $F_\sigma$ set in $\R$.
Set of Rational Numbers is not Closed in Reals
Let $\Q$ be the set of rational numbers. Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Then $\Q$ is not closed in $\R$.
Set of Rational Numbers is not G-Delta Set in Reals
Let $\Q$ be the set of rational numbers. Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Then $\Q$ is not a $G_\delta$ set in $\R$.
Eigenvalue of Matrix Powers
Let $A$ be a square matrix. Let $\lambda$ be an eigenvalue of $A$ and $\mathbf v$ be the corresponding eigenvector. Then: :$A^n \mathbf v = \lambda^n \mathbf v$ holds for each positive integer $n$.
Closure of Intersection of Rationals and Irrationals is Empty Set
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Let $\Q$ be the set of rational numbers. Then: :$\paren {\Q \cap \paren {\R \setminus \Q} }^- = \O$ where: :$\R \setminus \Q$ denotes the set of irrational numbers :$\paren {\Q \cap \paren {\R \setminus \Q} }^-$ denotes the closure of $\Q \cap \paren {\R \setminus \Q}$.
Intersection of Closures of Rationals and Irrationals is Reals
Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ be the set of rational numbers. Then: :$\Q^- \cap \paren {\R \setminus \Q}^- = \R$ where: :$\R \setminus \Q$ denotes the set of irrational numbers :$\Q^-$ denotes the closure of $\Q$.
Irrational Numbers form Metric Space
Let $\mathbb I = \R \setminus \Q$ be the set of all irrational numbers. Let $d: \mathbb I \times \mathbb I \to \R$ be defined as: :$\map d {x_1, x_2} = \size {x_1 - x_2}$ where $\size x$ is the absolute value of $x$. Then $d$ is a metric on $\mathbb I$ and so $\struct {\mathbb I, d}$ is a metric space.
Rational Number Space is Completely Normal
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is a completely normal space.
Irrational Number Space is Completely Normal
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is a completely normal space.
Rational Number Space is Paracompact
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is paracompact.
Euclidean Plus Metric is Metric
Let $\R$ be the set of real numbers. Let $d: \R \times \R \to \R$ be the Euclidean plus metric: :$\map d {x, y} := \size {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j}} - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Then $d$ is indeed a metric.
Open Ball in Euclidean Plus Metric is Subset of Equivalent Ball in Euclidean Metric
Let $\R$ be the set of real numbers. Let $d: \R \times \R \to \R$ be the Euclidean plus metric: :$\map d {x, y} := \size {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Let $d': \R \times \R \to \R$ be the Euclidean metric. Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number. Let $\map {B_\epsilon} {p; d}$ be an open $\epsilon$-ball of $p$ in $\R$ on $d$. Let $\map {B'_\epsilon} {p; d'}$ be an open $\epsilon$-ball of $p$ in $\R$ on $d'$. Then: :$\map {B_\epsilon} {p; d} \subseteq \map {B'_\epsilon} {p; d'}$