hcju/proofwikips huggingface dataset:
Basis Test for Limit Point
Let $T = \struct {S, \tau}$ be a topological space. Let $\BB$ be a synthetic basis of $T$. Let $H \subseteq S$. Then $x \in S$ is a limit point of $H$ : :$\forall U \in \BB : x \in U$ satisfies $H \cap U \setminus \set x \ne \O$
Basis Test for Adherent Point
Let $T = \struct {S, \tau}$ be a topological space. Let $\BB$ be a synthetic basis of $T$. Let $H \subseteq S$. Then $x \in S$ is an adherent point of $H$ : :$\forall U \in \BB : x \in U$ satisfies $H \cap U \ne \O$
Set of Liouville Numbers is Uncountable
The set of Liouville numbers is uncountable.
Isolated Point in Metric Space iff Isolated Point in Topological Space
Let $M = \struct {A, d}$ be a metric space. Let $T = \struct {A, \tau}$ be the topological space with the topology induced by $d$. Let $H \subseteq A$. Let $x \in H$ Then: :$x$ is an isolated point of $H$ in $M$  $x$ is an isolated point of $H$ in $T$
Limit Point in Metric Space iff Limit Point in Topological Space
Let $M = \struct {A, d}$ be a metric space. Let $T = \struct {A, \tau}$ be the topological space with the topology induced by $d$. Let $H \subseteq A$. Then: :$x \in H$ is a limit point in $M$  $x$ is a limit point in $T$
Sum of Functions of Bounded Variation is of Bounded Variation
Let $a, b$ be real numbers with $a < b$. Let $f, g : \closedint a b \to \R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f + g$ is of bounded variation with: :$V_{f + g} \le V_f + V_g$ where $V_{f + g}$ denotes the total variation of $f + g$.
Open Sets in Vector Spaces with Equivalent Norms Coincide
Let $M_a = \struct {X, \norm {\, \cdot \, }_a}$ and $M_b = \struct {X, \norm {\, \cdot \,}_b}$ be normed vector spaces. Let $U \subseteq X$ be an open set in $M_a$. Suppose, $\norm {\, \cdot \, }_a$ and $\norm {\, \cdot \,}_b$ are equivalent norms, i.e. $\norm {\, \cdot \, }_a \sim \norm {\, \cdot \,}_b$. Then $U$ is also open in $M_b$.
Local Basis Test for Isolated Point
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $x \in H$. Let $\BB_x$ be a local basis of $x$. Then $x$ is an isolated point of $H$ : :$\exists U \in \BB_x : U \cap H = \set x$
Local Basis Test for Limit Point
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $x \in S$. Let $\BB_x$ be a local basis of $x$. Then $x \in S$ is a limit point of $H$ : :$\forall U \in \BB_x : H \cap U \setminus \set x \ne \O$
Local Basis Test for Adherent Point
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $x \in S$. Let $\BB_x$ be a local basis of $x$. Then $x \in S$ is an adherent point of $H$ : :$\forall U \in \BB_x : H \cap U \ne \O$
Closure of Subset of Metric Space is Closed
Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$ be a subset of $A$. Let $H^-$ denote the closure of $H$. Then $H^-$ is a closed set of $M$.
Convergent Sequences in Vector Spaces with Equivalent Norms Coincide
Let $M_a = \struct {X, \norm {\, \cdot \, }_a}$ and $M_b = \struct {X, \norm {\, \cdot \,}_b}$ be normed vector spaces. Let $\sequence {x_n}_{n \mathop \in \N}$ be an convergent sequence in $M_a$. Suppose, $\norm {\, \cdot \, }_a$ and $\norm {\, \cdot \,}_b$ are equivalent norms, i.e. $\norm {\, \cdot \, }_a \sim \norm {\, \cdot \,}_b$. Then $\sequence {x_n}_{n \mathop \in \N}$ is also convergent in $M_b$.
Open Mapping is not necessarily Closed Mapping
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: T_1 \to T_2$ be a mapping which is not a bijection. Let $f$ be an open mapping. Then it is not necessarily the case that $f$ is also a closed mapping.
Equivalence of Definitions of Limit Point of Filter Basis
{{TFAE|def = Limit Point of Filter Basis}} Let $T = \struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on the underlying set $S$ of $T$. Let $\BB$ be a filter basis of $\FF$.
Richert's Theorem
Let $S = \set {s_1, s_2, \dots}$ be an infinite set of (strictly) positive integers, with the property: :$s_n < s_{n + 1}$ for every $n \in \N$ Suppose there exists some integers $N, k$ such that every integer $n$ with $N < n \le N + s_{k + 1}$: :$n$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_k}$ :$s_{i + 1} \le 2 s_i$ for every $i \ge k$ Then for any $n > N$, $n$ can be expressed as a sum of distinct elements in $S$.
Number as Sum of Distinct Primes
For $n \ne 1, 4, 6$, $n$ can be expressed as the sum of distinct primes.
Differentiable Function of Bounded Variation may not have Bounded Derivative
Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a continuous function of bounded variation. Let $f$ be differentiable on $\openint a b$. Then $f'$ is not necessarily bounded.
Existence of Urysohn Function does not guarantee Normal Space
Let $T = \struct {S, \tau}$ be a regular space. Let $T$ have the property that: :For all closed sets $A, B \subseteq S$ of $T$ such that $A \cap B = \O$, there exists an Urysohn function for $A$ and $B$. Then it is not necessarily the case that $T$ is a normal space.
Existence of Compact Space which Satisfies No Separation Axioms
There exists at least one example of a compact space for which none of the Tychonoff separation axioms are satisfied.
Everywhere Dense iff Interior of Complement is Empty
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subset S$. Then $A$ is everywhere dense : :$\paren {\relcomp S A}^\circ = \O$ where $A^\circ$ is the interior of $A$.
Greatest Set is Unique
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. Then the greatest set of $\TT$, if it exists, must be unique.
Greatest Set may not Exist
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. The greatest set of $\TT$ may not exist.
Smallest Set is Unique
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. Then the smallest set of $\TT$, if it exists, must be unique.
Smallest Set may not Exist
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. The smallest set of $\TT$ may not exist.
Mapping is Surjection if its Direct Image Mapping is Surjection
Let $f: S \to T$ be a mapping. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$. Let $f^\to$ be a surjection. Then $f: S \to T$ is also a surjection.
Nested Sequences in Complete Metric Space not Tending to Zero may be Disjoint
Let $M = \struct {A, d}$ be a complete metric space. Let $\family {S_k}_{k \mathop \in \N}$ be a nested sequence of closed balls in $M$. Let the radii of $\family {S_k}_{k \mathop \in \N}$ be convergent in $M$, but not to zero. Then it is not necessarily the case that their intersection $\displaystyle \bigcap S_k$ is non-empty.
Union of Interiors is Subset of Interior of Union
Let $T$ be a topological space. Let $\H$ be a set of subsets of $T$. That is, let $\H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$. Then the union of the interiors of the elements of $\H$ is a subset of the interior of the union of $\H$. :$\displaystyle \bigcup_{H \mathop \in \H} H^\circ \subseteq \paren {\bigcup_{H \mathop \in \H} H}^\circ $
Closed Ball is Closed/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x \in X$. Let $\epsilon \in \R_{> 0}$. Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$. Then $\map {B_\epsilon^-} x$ is a closed set of $M$.
Unit Sphere is Closed/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\Bbb S := \set {x \in X : \norm {x} = 1}$ be a unit sphere in $M$. Then $\Bbb S$ is closed in $M$.
Continued Fraction Expansion of Euler's Number/Proof 1/Lemma
:For $n \in \Z , n \ge 0$: {{begin-eqn}} {{eqn | l = A_n       | r = q_{3 n} e - p_{3 n} }} {{eqn | l = B_n       | r = p_{3 n + 1} - q_{3 n + 1} e }} {{eqn | l = C_n       | r = p_{3 n + 2} - q_{3 n + 2} e }} {{end-eqn}}
Limit Points in Particular Point Space/Subset
Let $U \subseteq S$ such that $p \in U$. Let $x \in S$ such that $x \ne p$. Then $x$ is a limit point of $U$.
Limit Points in Closed Extension Space/Subset
Let $U \subseteq S^*_p$ such that $p \in U$. Let $x \in S$. Then $x$ is a limit point of $U$.
Convergent Sequence in Particular Point Space
Let $T = \struct {S, \tau_p}$ be a particular point space. Let $\sequence {a_i}$ be a convergent sequence in $T$. Except for a finite number of indices, the terms of $\sequence {a_i}$ for which $a_i \ne p$ are all equal.
Accumulation Points for Sequence in Particular Point Space
Let $T = \struct {S, \tau_p}$ be a particular point space. Let $\sequence {a_i}$ be an infinite sequence in $T$. Let $\beta$ be an accumulation point of $\sequence {a_i}$. Then $\beta$ is such that an infinite number of terms of $\sequence {a_i}$ are equal either to $\beta$ or to $p$.
Absolute Value of Absolutely Continuous Function is Absolutely Continuous
Let $I \subseteq \R$ be a real interval. Let $f : I \to \R$ be an absolutely continuous function. Then $\size f$ is absolutely continuous.
Limit Points in Particular Point Space/Subset/Proof 2
Let $T = \struct {S, \tau_p}$ be a particular point space. {{:Limit Points in Particular Point Space/Subset}}
Limit Points in Particular Point Space/Subset/Proof 1
Let $T = \struct {S, \tau_p}$ be a particular point space. {{:Limit Points in Particular Point Space/Subset}}
Sum of Absolutely Continuous Functions is Absolutely Continuous
Let $I \subseteq \R$ be a real interval. Let $f, g : I \to \R$ be absolutely continuous functions. Then $f + g$ is absolutely continuous.
Norms on Finite-Dimensional Real Vector Space are Equivalent
Norms on finite-dimensional real vector space are equivalent.
Integral of Distribution Function
Let $\struct {X, \Sigma, \mu}$ be a measure space and $f$ be a $\mu$-measurable function. Let $p > 0, r \geq 0$. For $\lambda > 0$, let $E_\lambda = \set {x \in X: \size {\map f x} > \lambda}$, so that $\map m \lambda = \map \mu {E_\lambda}$ is the distribution function of $f$. Then: :$\displaystyle \int_0^\infty p \lambda^{p - 1} \int_{E_\lambda} \size f^r \rd \mu \rd \lambda = \int_X \size f^{p + r} \rd \mu$ and in particular: :$\displaystyle \int_0^\infty p \lambda^{p - 1} \map m \lambda \rd \lambda = \int_X \size f^p \rd \mu$
Either-Or Topology is Compact
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a compact space.
Compact Space is Lindelöf
Every compact space is Lindelöf.
Limit Point of Underlying Set of Sequence of Reciprocals and Reciprocals + 1
Let $\sequence {a_n}$ denote the sequence defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $S$ denote the set of terms of $\sequence {a_n}$ considered as a subset of $\struct {\R, \tau_d}$. Then $0$ is a limit point of $S$.
Omega-Accumulation Point of Underlying Set of Sequence of Reciprocals and Reciprocals + 1
Let $\sequence {a_n}$ denote the sequence defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $S$ denote the set of terms of $\sequence {a_n}$ considered as a subset of $\struct {\R, \tau_d}$. Then $0$ is an $\omega$-accumulation point of $S$.
Accumulation Point of Sequence of Reciprocals and Reciprocals + 1
Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $\sequence {a_n}$ denote the sequence in $\struct {\R, \tau}$ defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Then $0$ is an accumulation point of $\sequence {a_n}$.
Zero is not a Limit Point of Sequence of Reciprocals and Reciprocals + 1
Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $\sequence {a_n}$ denote the sequence in $\struct {\R, \tau}$ defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Then $0$ is not a limit point of $\sequence {a_n}$.
1 plus Power of 2 is not Perfect Power except 9
The only solution to: :$1 + 2^n = a^b$ is: :$\tuple {n, a, b} = \tuple {3, 3, 2}$ for positive integers $n, a, b$ with $b > 1$.
Fermat Number is not Perfect Power
There exist no Fermat numbers which are perfect powers.
Uncountable Closed Ordinal Space is Countably Compact
Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is a countably compact space.
Sum of Euler Numbers by Binomial Coefficients Vanishes
$\forall n \in \Z_{>0}: \displaystyle \sum_{k \mathop = 0}^n \binom {2 n} {2 k} E_{2 k} = 0$ where $E_k$ denotes the $k$th Euler number. == Corollary == Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = -\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}       | c =  }} {{eqn | r = -\paren {\binom {2 n} 0 E_0 + \binom {2 n} 2 E_2 + \binom {2 n} 4 E_4 + \cdots + \binom {2 n} {2 n - 2} E_{2 n - 2} }       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number.
Uncountable Open Ordinal Space is Countably Compact
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is a countably compact space.
Uncountable Open Ordinal Space is not Metacompact
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a metacompact space.
Uncountable Open Ordinal Space is not Paracompact
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a paracompact space.
Uncountable Open Ordinal Space is not Lindelöf
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a Lindelöf space.
Uncountable Open Ordinal Space is not Sigma-Compact
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a $\sigma$-compact space.
Uncountable Closed Ordinal Space is Lindelöf
Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is a Lindelöf space.
Uncountable Open Ordinal Space is Sequentially Compact
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is a sequentially compact space.
Sum of Euler Numbers by Binomial Coefficients Vanishes/Corollary
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = -\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}       | c =  }} {{eqn | r = -\paren {\binom {2 n} 0 E_0 + \binom {2 n} 2 E_2 + \binom {2 n} 4 E_4 + \cdots + \binom {2 n} {2 n - 2} E_{2 n - 2} }       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number.
Integer to Power of Multiple of Order/Corollary
Then $\map \phi n$ is a multiple of $c$, where $\map \phi n$ is the Euler phi function of $n$.
Divisor of Fermat Number/Euler's Result
Then $m$ is in the form: :$k \, 2^{n + 1} + 1$ where $k \in \Z_{>0}$ is an integer.
Divisor of Fermat Number/Refinement by Lucas
Let $n \ge 2$. Then $m$ is in the form: :$k \, 2^{n + 2} + 1$
Definition:Double Pointed Real Number Line
Let $T_\R = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$. Let $T = T_\R \times T_D$ be theproduct space of $T_\R$ and $T_D$. $T$ is known as the '''double pointed real number line'''.
Set Closure is Smallest Closed Set/Normed Vector Space
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $S$ be a subset of $X$: :$S \subseteq X$ Let $S^-$ be the closure of $S$. Then $S^-$ is the smallest closed set which contains $S$.
Bounded Real Function may not be of Bounded Variation
Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a bounded function. Then $f$ is not necessarily of bounded variation.
Element is Loop iff Member of Closure of Empty Set
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $x \in S$. Then: :$x$ is a loop  $x \in \map \sigma \O$ where $\map \sigma \O$ denotes the closure of the empty set.
Singleton is Dependent implies Rank is Zero/Corollary
:$x$ is a loop  $\map \rho {\set x} = 0$
Superset of Dependent Set is Dependent/Corollary
Let $A \subseteq S$. Let $x \in A$. If $x$ is a loop then $A$ is dependent.
Closure of Subset contains Loop
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $x$ be a loop of $M$. Let $A \subseteq S$. Then: :$x \in \map \sigma A$ where $\map \sigma A$ denotes the closure of $A$.
Element is Loop iff Singleton is Circuit
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $x \in S$. Then: :$x$ is a loop  $\set x$ is a circuit
Element is Member of Base iff Not Loop
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $\mathscr B$ denote the set of all bases of $M$. Let $x \in S$. Then: :$\exists B \in \mathscr B: x \in B$  $x$ is not a loop
Parallel Relationship is Transitive
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $x, y, z \in S : x \ne y, x \ne z, y \ne z$. If $x$ is parallel to $y$ and $y$ is parallel to $z$ then $x$ is parallel to $z$.
Distinct Matroid Elements are Parallel iff Each is in Closure of Other
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$. Let $x, y \in S : x \ne y$. Then $x$ is parallel to $y$ : :$(1)\quad x$ and $y$ are not loops :$(2)\quad x \in \map \sigma {\set y}$ :$(3)\quad y \in \map \sigma {\set x}$
Closure of Subset Contains Parallel Elements
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$. Let $A \subseteq S$. Let $x, y \in S$. If $x \in \map \sigma A$ and $y$ is parallel to $x$ then: :$y \in \map \sigma A$
Loop Belongs to Every Flat
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $x \in S$. If $x$ is a loop and $A$ is a flat subset then $x \in A$.
Parallel Elements Depend on Same Subsets
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $x, y \in S$. Let $x$ be parallel to $y$. Then: :$x$ depends on $A$  $y$ depends on $A$
Absolutely Continuous Real Function is Uniformly Continuous
Let $I \subseteq \R$ be a real interval. Let $f : I \to \R$ be an absolutely continuous function. Then $f$ is uniformly continuous.
Closure of Convex Subset in Normed Vector Space is Convex
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $C \subseteq X$ be a convex subset of $X$. Let $C^-$ be the closure of $C$. Then $C^- \subseteq X$ is also a convex subset of $X$.
Minimal Number of Distinct Prime Factors for Integer to have Abundancy Index Exceed Given Number
Let $r \in \R$. Let $\mathbb P^-$ be the set of prime numbers with possibly finitely many numbers removed. Define: :$M = \min \set {m \in \N: \displaystyle \prod_{i \mathop = 1}^m \frac {p_i} {p_i - 1} > r}$ where $p_i$ is the $i$th element of $\mathbb P^-$, ordered by size. Then $M$ satisfies: :$(1): \quad$ Every number formed with fewer than $M$ distinct prime factors in $\mathbb P^-$ has abundancy index less than $r$ :$(2): \quad$ There exists some number formed with $M$ distinct prime factors in $\mathbb P^-$ with abundancy index at least $r$ So $M$ is the minimal number of distinct prime factors in $\mathbb P^-$ a number must have for it to have abundancy index at least $r$. For $r$ an integer greater than $1$: If $\mathbb P^-$ is taken to be the set of all prime numbers, the values of $M$ are: :$2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, \cdots$ {{OEIS|A005579}} This theorem shows that this sequence is a subsequence of the sequence A256969 in the OEIS, only differing by an offset. If we require the numbers to be odd, we remove $2$ from $\mathbb P^-$. The sequence of values of $M$ are: :$3, 8, 21, 54, 141, 372, 995, 2697, 7397, 20502, \cdots$ {{OEIS|A005580}}
1 plus Square is not Perfect Power
The equation: :$x^p = y^2 + 1$ has no solution in the integers for $x, y, p > 1$.
1 plus Perfect Power is not Prime Power except for 9
The only solution to: :$x^m = y^n + 1$ is: :$\tuple {x, m, y, n} = \tuple {3, 2, 2, 3}$ for positive integers $x, y, m, n > 1$, and $x$ is a prime number. This is a special case of Catalan's Conjecture.
Field is Principal Ideal Domain
Let $F$ be a field. Then $F$ is a principal ideal domain.
Complex Vector Space is Vector Space
Let $\C$ denote the set of complex numbers. Then the complex vector space $\C^n$ is a vector space.
Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition
Let $\struct {X, \norm {\, \cdot \,}}$ is a normed vector space. Let $D \subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$.
Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition
Let $\struct {X, \norm {\, \cdot \,}}$ is a normed vector space. Let $D \subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$.
Modulus of Limit/Normed Vector Space
Let $\struct {X, \norm { \, \cdot \, } }$ be a normed vector space. Let $\sequence {x_n}$ be a convergent sequence in $R$ to the limit $x$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = x$. Then :$\displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = \norm x$
Matrix Entrywise Addition forms Abelian Group
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$. Then $\struct {\map {\MM_R} {m, n}, +}$, where $+$ is matrix entrywise addition, is a group.
Definition:Negative Matrix/General Group
Let $\struct {G, \cdot}$ be a group. Let $\map {\MM_G} {m, n}$ denote the $m \times n$ matrix space over $\struct {G, \cdot}$. Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\struct {\map {\MM_G} {m, n}, \circ}$, where $\circ$ is the Hadamard product. Then the '''negative (matrix) of $\mathbf A = \sqbrk a_{m n}$''' is denoted and defined as: :$-\mathbf A := \sqbrk {a^{-1} }_{m n}$ where $a^{-1}$ is the inverse element of $a \in G$.
Negative Matrix is Inverse for Matrix Entrywise Addition over Ring
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$. Let $\mathbf A$ be an element of $\map {\MM_R} {m, n}$. Let $-\mathbf A$ be the negative of $\mathbf A$. Then $-\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition.
Zero Matrix is Identity for Matrix Entrywise Addition over Ring
Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the zero matrix of $\map {\MM_R} {m, n}$. Then $\mathbf 0_R$ is the identity element for matrix entrywise addition.