hcju/proofwikips huggingface dataset:
Uniqueness of Polynomial Ring in One Variable
Let $R$ be a commutative ring with unity. Let $\struct {R \sqbrk X, \iota, X}$ and $\struct {R \sqbrk Y, \kappa, Y}$ be polynomial rings in one variable over $R$. Then there exists a unique ring homomorphism $f: R \sqbrk X \to R \sqbrk Y$ such that: :$f \circ \iota = \kappa$ :$\map f X = Y$ and it is an isomorphism.
Monomials of Polynomial Ring are Linearly Independent/One Variable
Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Then the set of monomials $\set {X^k : k \in \N}$ is linearly independent.
Equivalence of Definitions of Constant Polynomial
Let $R$ be a commutative ring with unity. Let $P\in R[x]$ be a polynomial in one variable over $R$. {{TFAE|def = Constant Polynomial}}
Algebraic Element of Field Extension is Root of Unique Monic Polynomial of Minimal Degree
Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Then there exists a unique monic polynomial $f \in K \sqbrk x$ of smallest degree such that $\map f \alpha = 0$, called the '''minimal polynomial'''.
Algebraic Element of Field Extension is Root of Unique Monic Irreducible Polynomial
Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Then there exists a unique irreducible monic polynomial $f\in K[x]$ such that $f(\alpha) = 0$, called the '''minimal polynomial'''.
Annihilating Polynomial of Minimal Degree is Irreducible
Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Let $f \in K[x]$ be a nonzero polynomial of minimal degree such that $\map f \alpha = 0$. Then $f$ is irreducible in $K \sqbrk x$.
Nonzero Ideal of Polynomial Ring over Field has Unique Monic Generator
Let $K$ be a field. Let $K \sqbrk x$ be the polynomial ring in one variable over $K$. Let $I \subseteq K \sqbrk x$ be a nonzero ideal. Then $I$ is generated by a unique monic polynomial.
Equivalence of Definitions of Integral Element of Algebra
{{TFAE|def = Integral Element of Algebra}} Let $A$ be a commutative ring with unity. Let $f : A \to B$ be a commutative $A$-algebra. Let $b\in B$.
Product of Closed Sets is Closed
Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary indexing set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$. Let $T = \struct {S, \TT}$ be the product space of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ with Tychonoff topology $\TT$. Suppose we have an indexed family of sets $\family {C_i}_{i \mathop \in I}$, where each $C_i$ is closed in $\struct {S_i, \tau_i}$. Then $\displaystyle \prod_{i \mathop \in I} C_i$ is closed in $\struct {S, \TT}$.
Prime Decomposition of 9th Fermat Number
The prime decomposition of the $9$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\left({2^9}\right)} + 1       | r = 13 \, 407 \, 807 \, 929 \, 942 \, 597 \, 099 \, 574 \, 024 \, 998 \, 205 \, 846 \, 127 \, 479 \, 365 \, 820 \, 592 \, 393 \, 377 \, 723 \, 561 \, 443 \, 721 \, 764 \, 030 \, 073 \, 546 \, 976 \, 801 \, 874 \, 298 \, 166 \, 903 \, 427 \, 690 \, 031 \, 858 \, 186 \, 486 \, 050 \, 853 \, 753 \, 882 \, 811 \, 946 \, 569 \, 946 \, 433 \, 649 \, 006 \, 084 \, 097       | c =  }} {{eqn | r = 2 \, 424 \, 833       | c =  }} {{eqn | o =       | ro= \times       | r = 7 \, 455 \, 602 \, 825 \, 647 \, 884 \, 208 \, 337 \, 395 \, 736 \, 200 \, 454 \, 918 \, 783 \, 366 \, 342 \, 657       | c =  }} {{eqn | o =       | ro= \times       | r = 741 \, 640 \, 062 \, 627 \, 530 \, 801 \, 524 \, 787 \, 141 \, 901 \, 937 \, 474 \, 059 \, 940 \, 781 \, 097 \, 519 \, 023 \, 905 \, 821 \, 316 \, 144 \, 415 \, 759 \, 504 \, 705 \, 008 \, 092 \, 818 \, 711 \, 693 \, 940 \, 737       | c =  }} {{eqn | r = \paren {2^5 \times 37 \times 2^{11} + 1}       | c =  }} {{eqn | o =       | ro= \times       | r = \paren {19 \times 47 \times 82 \, 488 \, 781 \times 1 \, 143 \, 290 \, 228 \, 161 \, 321 \times 43 \, 226 \, 490 \, 359 \, 557 \, 706 \, 629 \times 2^{11} + 1}       | c =  }} {{eqn | o =       | ro= \times       | r = \paren {1129 \times 26 \, 813 \times 40 \, 644 \, 377 \times 17 \, 338 \, 437 \, 577 \, 121 \times 16 \, 975 \, 143 \, 302 \, 271 \, 505 \, 426 \, 897 \, 585 \, 653 \, 131 \, 126 \, 520 \, 182 \, 328 \, 037 \, 821 \, 729 \, 720 \, 833 \, 840 \, 187 \, 223 \times 2^{11} + 1}       | c =  }} {{end-eqn}}
Prime Number Formed by Concatenating Consecutive Integers down to 1
Let $N$ be an integer whose decimal representation consists of the concatenation of all the integers from a given $n$ in descending order down to $1$. Let the $N$ that is so formed be prime. The only $n$ less than $100$ for which this is true is $82$. That is: :$82 \, 818 \, 079 \, 787 \, 776 \ldots 121 \, 110 \, 987 \, 654 \, 321$ is the only prime number formed this way starting at $100$ or less.
Yoneda Lemma for Covariant Functors
Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. === Bijection === {{:Bijection in Yoneda Lemma for Covariant Functors}} === Naturality === {{:Naturality of Yoneda Lemma for Covariant Functors}}
Naturality of Yoneda Lemma for Covariant Functors
Let $[C, \mathbf{Set}]$ be the covariant functor category. Let $C \times [C, \mathbf{Set}] $ be the product category. Let $C \times [C, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto \operatorname{Nat}(h^A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the contravariant Yoneda functor $h^-$ and the identity functor $\operatorname{id}_{[C, \mathbf{Set}]}$. Let $\operatorname{ev} : C \times [C, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto F(A)$ be the functor evaluation functor. Then $\Phi_{(A, F)} : \operatorname{Nat}(h^A, F) \to F(A) : \eta \mapsto \eta_A(\operatorname{id}_A)$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.
Alternating Even-Odd Digit Palindromic Prime
Let the notation $\left({abc}\right)_n$ be interpreted to mean $n$ consecutive repetitions of a string of digits $abc$ concatenated in the decimal representation of an integer. The integer: :$\left({10987654321234567890}\right)_{42} 1$ has the following properties: : it is a palindromic prime with $841$ digits : its digits are alternately odd and even.
Bijection in Yoneda Lemma for Covariant Functors
Let $F: C \to \mathbf {Set}$ be a covariant functor. Let $A \in C$ be an object. Let $I_A$ be its identity morphism. Let $h^A = \map {\operatorname {Hom} } {A, -}$ be its covariant hom-functor. The class of natural transformations $\map {\operatorname {Nat} } {h^A, F}$ is a small class, and: :$\alpha: \map {\operatorname {Nat} } {h^A, F} \to \map F A: \eta \mapsto \map {\eta_A} {I_A}$ :$\beta: \map F A \to \map {\operatorname {Nat} } {h^A, F}: u \mapsto \paren {X \mapsto \paren {f \mapsto \map {\paren {\map F f} } u} }$ are inverses of each other.
Yoneda Lemma for Contravariant Functors
Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. === Bijection === {{:Bijection in Yoneda Lemma for Contravariant Functors}} === Naturality === {{:Naturality of Yoneda Lemma for Contravariant Functors}}
Naturality of Yoneda Lemma for Contravariant Functors
Let $[C^{\operatorname{op}}, \mathbf{Set}]$ be the contravariant functor category. Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] $ be the product category. Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto \operatorname{Nat}(h_A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the opposite of the covariant Yoneda functor $h_-$ and the identity functor $\operatorname{id}_{[C^{\operatorname{op}}, \mathbf{Set}]}$. Let $\operatorname{ev} : C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto F(A)$ be the contravariant functor evaluation functor. Then $\Phi_{(A, F)} : \operatorname{Nat}(h_A, F) \to F(A) : \eta \mapsto \eta_A(\operatorname{id}_A)$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.
Period of Reciprocal of Repunit 1031 is 1031
The decimal expansion of the reciprocal of the repunit prime $R_{1031}$ has a period of $1031$. :$\dfrac 1 {R_{1031}} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{1030} \dot 9$ This is the only prime number to have a period of exactly $1031$.
Titanic Prime whose Digits are all Prime
The integer defined as: :$7532 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$ is a titanic prime all of whose digits are themselves prime. That is: :$\underbrace{7532}_{275} 7533$
Titanic Prime whose Digits are all 0 or 1
The integer defined as: :$10^{641} \times \dfrac{10^{640} - 1} 9 + 1$ is a titanic prime all of whose digits are either $0$ or $1$. That is: :$\left({1}\right)_{640} \left({0}\right)_{640} 1$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.
Titanic Prime whose Digits are all 9 except for one 1
The integer defined as: :$2 \times 10^{3020} - 1$ is a titanic prime all of whose digits are $9$ except one, which is $1$. That is: :$1 \left({9}\right)_{3020}$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.
Titanic Prime consisting of Blocks of 111 of each Digit plus Zeroes
The integer defined as: :$\left({1}\right)_{111} \left({2}\right)_{111} \left({3}\right)_{111} \left({4}\right)_{111} \left({5}\right)_{111} \left({6}\right)_{111} \left({7}\right)_{111} \left({8}\right)_{111} \left({9}\right)_{111} \left({0}\right)_{2284} 1$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string, is a titanic prime.
Pair of Titanic Twin Primes
The integers defined as: :$190 \, 116 \times 3003 \times 10^{5120} \pm 1$ are a pair of titanic twin primes. That is: :$570 \, 918 \, 347 \paren 9_{5820}$ and: :$570 \, 918 \, 348 \paren 0_{5819} 1$ where $\paren a_b$ means $b$ instances of $a$ in a string.
Indexed Iterated Operation does not Change under Permutation
Let $G$ be a commutative semigroup. Let $a, b \in\Z$ be integers. Let $\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \closedint a b \to G$ be a mapping. Let $\sigma: \closedint a b \to \closedint a b$ be a permutation.
Summation over Cartesian Product as Double Summation
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S, T$ be finite sets. Let $S \times T$ be their cartesian product. Then we have an equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{\paren {s, t} \mathop \in S \times T} \map f {s, t}$
Zeroth Hyperoperation is Successor Function
The '''zeroth hyperoperation''' is the successor function: :$H_0 \left({x, y}\right) = y + 1$
First Hyperoperation is Addition Operation
The '''$1$st hyperoperation''' is the addition operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_1 \left({x, y}\right) = x + y$
Second Hyperoperation is Multiplication Operation
The '''$2$nd hyperoperation''' is the multiplication operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_2 \left({x, y}\right) = x \times y$
Third Hyperoperation is Integer Power Operation
The '''$3$rd hyperoperation''' is the integer power operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_3 \left({x, y}\right) = x^y$
Fourth Hyperoperation is Tetration Operation
The '''$4$th hyperoperation''' is the tetration operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$ where $\uparrow \uparrow$ denotes tetration: :$x \uparrow \uparrow n := \begin{cases} 1 & : n = 0 \\ x \uparrow \left({x \uparrow \uparrow \left({n - 1}\right)}\right) & : n > 0 \\ \end{cases}$ In this context, $x \uparrow y$ is the Knuth notation for powers: :$x \uparrow y := x^y$
Largest Integer Expressible by 3 Digits
The largest integer that can be represented using no more than $3$ digits, with no additional symbols, is: :$9^{9^9} = 9^{387 \, 420 \, 489}$ and (at $369 \, 693 \, 100$ digits, is too large to be calculated on a conventional calculator. Note that this does not include the notation for tetration: ${}^9 9$.
Largest Integer Expressible by 3 Digits/Logarithm Base 10
:$\map {\log_{10} } {9^{9^9} } \approx 369 \, 693 \,099 \cdotp 63157 \, 03685 \, 87876 \, 1$
Largest Integer Expressible by 3 Digits/Number of Digits
:$9^{9^9}$ has $369 \, 693 \, 100$ digits when expressed in decimal notation.
Number of Primes up to n Approximates to Eulerian Logarithmic Integral
The prime-counting function approximates to the Eulerian logarithmic integral: :$\map \pi n \approx \displaystyle \int_2^n \frac {\d x} {\ln x}$
Point at which Prime-Counting Function becomes less than Eulerian Logarithmic Integral
Let $\map \pi n$ denote the prime-counting function. Let $a \uparrow b$ be interpreted as Knuth notation for $a^b$.
Reciprocals of Prime Numbers
The decimal representations of the reciprocals of the first few prime numbers are as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="left" style = "padding: 2px 10px" | $1 / n$  ! style = "padding: 2px 10px" | Also see |- | align="right" style = "padding: 2px 10px" | $2$ | align="left" style = "padding: 2px 10px" | $0 \cdotp 5$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $3$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 3$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $5$ | align="left" style = "padding: 2px 10px" | $0 \cdotp 2$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $7$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 14285 \, \dot 7$ | style = "padding: 2px 10px" | Period of Reciprocal of 7 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $11$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 0 \dot 9$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $13$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 07692 \dot 3$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $17$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ | style = "padding: 2px 10px" | Period of Reciprocal of 17 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $19$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ | style = "padding: 2px 10px" | Period of Reciprocal of 19 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $23$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ | style = "padding: 2px 10px" | Period of Reciprocal of 23 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $29$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $31$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$ | style = "padding: 2px 10px" | Period of Reciprocal of 31 is of Odd Length |- | align="right" style = "padding: 2px 10px" | $37$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 02 \dot 7$ | style = "padding: 2px 10px" | Period of Reciprocal of 37 has Length 3 |}
Factors of Repunits
The prime factors of the first few repunits are as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $R_n$  ! align="left" style = "padding: 2px 10px" | Factors of $n$  ! style = "padding: 2px 10px" | Also see |- | align="right" style = "padding: 2px 10px" | $2$ | align="left" style = "padding: 2px 10px" | $11$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $3$ | align="left" style = "padding: 2px 10px" | $3 \times 37$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $4$ | align="left" style = "padding: 2px 10px" | $11 \times 101$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $5$ | align="left" style = "padding: 2px 10px" | $41 \times 271$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $6$ | align="left" style = "padding: 2px 10px" | $3 \times 7 \times 11 \times 13 \times 37$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $7$ | align="left" style = "padding: 2px 10px" | $239 \times 4649$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $8$ | align="left" style = "padding: 2px 10px" | $11 \times 73 \times 101 \times 137$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $9$ | align="left" style = "padding: 2px 10px" | $3^2 \times 37 \times 333 \, 667$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $10$ | align="left" style = "padding: 2px 10px" | $11 \times 41 \times 271 \times 9091$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $11$ | align="left" style = "padding: 2px 10px" | $21 \, 649 \times 513 \, 239$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $12$ | align="left" style = "padding: 2px 10px" | $3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $13$ | align="left" style = "padding: 2px 10px" | $53 \times 79 \times 265 \, 371 \, 653$ | style = "padding: 2px 10px" |  |}
Separability is not Weakly Hereditary
The property of separability is not weakly hereditary.
Filter is Finer iff Sets of Basis are Subsets
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF, \FF' \subset \powerset S$ be two filters on $S$. Let $\FF$ have a basis $\BB$. Let $\FF'$ have a basis $\BB'$. $\FF$ is finer than $\FF'$  for every set of $\BB'$, there is a set of $\BB$ subset to it.
Existence of Topological Space which satisfies no Separation Axioms but T0
There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom.
Existence of Topological Space which satisfies no Separation Axioms but T0 and T1
There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom and $T_1$ (Fréchet) axiom.
Equivalence of Definitions of Unital Associative Commutative Algebra
Let $A$ be a commutative ring with unity.
Existence of Hausdorff Space which is not T3, T4 or T5
There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space (and hence also $T_0$ (Kolmogorov) and $T_1$ (Fréchet), but is not a $T_3$ space, a $T_4$ space or a $T_5$ space.
Existence of Topological Space which satisfies no Separation Axioms but T3
There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_3$ axiom.
Existence of Topological Space which satisfies no Separation Axioms but T4
There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_4$ axiom.
Existence of Tychonoff Space which is not Normal
There exists at least one example of a topological space which is a Tychonoff space, but is not also a normal space.
Existence of T4 Space which is not T3 1/2
There exists at least one example of a topological space which is a $T_4$ space, but is not also a $T_{3 \frac 1 2}$ space.
Existence of Normal Space which is not Completely Normal
There exists at least one example of a topological space which is a normal space, but is not also a completely normal space.
Separation Properties Not Preserved by Expansion
These separation properties are not generally preserved under expansion: :$T_3$ Space :Regular Space :$T_4$ Space :Completely Regular Space :$T_5$ Space :Normal Space :Completely Normal Space
Existence of Completely Normal Space whose Product Space is Not Normal
There exists at least one example of a completely normal topological space $T$ such that the product space $T \times T$ is not a normal topological space.
Existence of Completely Normal Space which is not Perfectly Normal
There exists at least one example of a completely normal topological space which is not perfectly normal.
Existence of Semiregular Topological Space which is not T3
There exists at least one example of a semiregular topological space which is not a $T_3$ space.
Existence of Semiregular Topological Space which is not Completely Hausdorff
There exists at least one example of a semiregular topological space which is not a completely Hausdorff space.
Existence of Sigma-Compact Space which is not Compact
There exists at least one example of a $\sigma$-compact topological space which is not also a compact space.
Existence of Lindelöf Space which is not Sigma-Compact
There exists at least one example of a $\sigma$-compact topological space which is not also a compact space.
Existence of Countably Compact Space which is not Sequentially Compact
There exists at least one example of a countably compact topological space which is not also a sequentially compact space.
Existence of Weakly Countably Compact Space which is not Countably Compact
There exists at least one example of a weakly countably compact topological space which is not also a countably compact space.
Existence of Pseudocompact Space which is not Countably Compact
There exists at least one example of a pseudocompact topological space which is not also a countably compact space.
Vectorization of Product of Three Matrices
Let $R$ be a ring. Let $A, B, C$ be matrices over $R$ such that the matrix product $ABC$ is defined. Then $\map {\operatorname {vec} }{ABC} = \paren {C^\intercal \otimes A} \cdot \map {\operatorname {vec} } B$ where: :$\operatorname {vec}$ denotes vectorization :$C^\intercal$ is the transpose of $C$ :$\otimes$ denotes Kronecker product :$\cdot$ denotes matrix product
Existence of Weakly Sigma-Locally Compact Space which is not Strongly Locally Compact
There exists at least one example of a weakly $\sigma$-locally compact topological space which is not also a strongly locally compact space.
Existence of First-Countable Space which is not Second-Countable
There exists at least one example of a first-countable topological space which is not also a second-countable space.
Existence of Lindelöf Space which is not Second-Countable
There exists at least one example of a Lindelöf space which is not also a second-countable space.
Existence of Topological Space satisfying Countable Chain Condition which is not Separable
There exists at least one example of a topological space which satisfies the countable chain condition which is not also a separable space.
Equivalence of Definitions of Strongly Locally Compact Space
Let $T = \left({S, \tau}\right)$ be a topological space. {{TFAE|def = Strongly Locally Compact Space}}
Uniqueness of Representing Objects
Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor. Let $(A, \eta)$ and $(B, \xi)$ be representations of $F$. Then there exists a unique isomorphism $f : A \to B$ such that $\eta \circ h^f = \xi$, where: :$h^f$ is the precomposition natural transformation :$\circ$ denotes vertical composition of natural transformations
Existence and Uniqueness of Reduced Form of Group Word
Let $X$ be a set. Let $w$ be a group word on $X$. Then $w$ has a unique reduced form.
Existence of Metacompact Space which is not Paracompact
There exists at least one example of a metacompact topological space which is not also a paracompact space.
Existence of Minimal Hausdorff Space which is not Compact
Let $S$ be a set. Let $\tau$ be the minimal subset of the power set $\powerset S$ such that $\struct {S, \tau}$ is a Hausdorff space. Then it is not necessarily the case that $\struct {S, \tau}$ is compact.