hcju/proofwikips huggingface dataset:
Filters of Lattice of Power Set form Bounded Below Ordered Set
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an inclusion lattice of power set of $X$. Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an inclusion ordered set, where $\mathit{Filt}\left({L}\right)$ denotes the set of all filters on $L$. Then $F$ is bounded below and $\bot_F = \left\{{X}\right\}$ where $\bot_F$ denotes the smallest element of $F$.
Deterministic Time Hierarchy Theorem
Let $\map f n$ be a time-constructible function. Then there exists a decision problem which: :can be solved in worst-case deterministic time $\map f {2 n + 1}^3$ but: :cannot be solved in worst-case deterministic time $\map f n$. In other words, the complexity class $\map {\mathsf {DTIME} } {\map f n} \subsetneq \map {\mathsf {DTIME} } {\map f {2 n + 1}^3}$.
Points in Product Spaces are Near Open Sets
Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of topological spaces, where $I$ is an arbitrary index set. Let $X = \displaystyle \prod_{i \mathop \in I} X_i$ be the product space of $\family {X_i}_{i \mathop \in I}$. Let $U$ be nonempty open subset of $X$. Let $x$ be a point in $X$. For each point $y$ in $X$, let $\map K y = \set {i \in I : y_i \ne x_i}$. Then there exists a point $u$ in $U$ such that $\map K u$ is finite.
Equivalence of Definitions of Limit of Function in Metric Space
{{TFAE|def = Limit of Function (Metric Space)|view = Limit of Function|context = Metric Space|contextview = Metric Spaces}} Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $L \in M_2$.
Equivalence of Definitions of Field of Quotients
Let $D$ be an integral domain. Let $F$ be a field. {{TFAE|def = Field of Quotients}}
Products of 2-Digit Pairs which Reversed reveal Same Product
The following positive integers can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other: :$504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$ <!-- fascists won't include the damn thing {{OEIS|A289978}}  SIEG HEIL! SIEG HEIL!-->
Poulet Numbers which are also Magic Constant for Magic Square
The sequence of Poulet numbers which are also the magic constant of a magic square begins: :$1105, 2465, \ldots$
Length of Contour is Well-Defined
Let $C_1, \ldots, C_n$ be directed smooth curves. Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. Suppose that $\sigma_i: \closedint {c_i} {d_i} \to \C$ is a reparameterization of $C_i$ for all $i \in \set {1, \ldots, n}$ Then: :$\displaystyle \sum_{i \mathop = 1}^n \int_{a_i }^{b_i} \size {\map {\gamma_i'} t} \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} \size {\map {\sigma_i'} t} \rd t$ and all real integrals in the equation are defined.
Equivalence of Definitions of Change of Basis Matrix
Let $R$ be a ring with unity. Let $G$ be a finite-dimensional unitary $R$-module. Let $A = \sequence {a_n}$ and $B = \sequence {b_n}$ be ordered bases of $G$. {{TFAE|def = Change of Basis Matrix}}
Numbers not Expressible as Sum of Fewer than 19 Fourth Powers
The following positive integer are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers: :$79, 159, 239, 319, 399, 479, 559$ {{OEIS|A046050}}
Free Module on Set is Free
Let $R$ be a ring with unity. Let $I$ be a set. Let $R^{\paren I}$ be the free $R$-module on $I$. Then $R^{\paren I}$ is a free $R$-module.
Pandigital Square Equation
The following equations, which include each digit from $1$ to $9$ inclusive, are the only ones of their kind: {{begin-eqn}} {{eqn | l = 567^2       | r = 321 \, 489 }} {{eqn | l = 854^2       | r = 729 \, 316 }} {{end-eqn}}
Sequence of 11 Primes by Trebling and Adding 16
The process of multiplication by $3$ and then adding $16$ produces a sequence of $11$ primes when starting from $587$: :$587, 1777, 5347, 16 \, 057, 48 \, 187, 144 \, 577, 433 \, 747, 1 \, 301 \, 257, 3 \, 903 \, 787, 11 \, 711 \, 377, 35 \, 134 \, 147$
Consecutive Sophie Germain Primes cannot be Pair of Twin Primes
Let $p$ and $p + 2$ be twin primes. Then unless $p = 3$ it is not possible for both $p$ and $p + 2$ to be Sophie Germain primes.
Ring Without Unity may have Quotient Ring with Unity
Let $\struct {R, +, \circ}$ be a ring. Let $I$ be an ideal of $R$. Let $\struct {R / I, +, \circ}$ be the associated quotient ring. Then $\struct {R / I, +, \circ}$ may have a unity even if $\struct {R, +, \circ}$ has not.
Stone Space is Topological Space
Let $\struct {B, \preceq, \wedge, \vee}$ be a non-empty Boolean algebra. Let $\struct {U, \tau}$ be the Stone space of $B$. Then $\struct {U, \tau}$ is a topological space.
Sum of 714 and 715
The sum of $714$ and $715$ is a $4$-digit integer which has $6$ anagrams which are prime.
Limits of Real and Imaginary Parts
Let $f: D \to \C$ be a complex function, where $D \subseteq \C$. Let $z_o \in D$ be a complex number. Suppose $f$ is continuous at $z_0$. Then: :$(1): \quad \displaystyle \lim_{z \to z_o} \operatorname{Re} \left({f \left({z}\right) }\right) = \operatorname{Re} \left({ \lim_{z \to z_o} f \left({z}\right) }\right)$ :$(2): \quad \displaystyle \lim_{z \to z_o} \operatorname{Im} \left({f \left({z}\right) }\right) = \operatorname{Im} \left({ \lim_{z \to z_o} f \left({z}\right) }\right)$ Here, $\operatorname{Re} \left({f \left({z}\right) }\right) $ denotes the real part of $f \left({z}\right)$, and $\operatorname{Im} \left({f \left({z}\right) }\right) $ denotes the imaginary part of $f \left({z}\right)$.
Equivalence of Definitions of Synthetic Basis
Let $S$ be a set. {{TFAE|def = Synthetic Basis}}
Continuous iff Mapping at Element is Supremum of Compact Elements
Let $L = \left({S, \preceq_1, \tau_1}\right)$ and $R = \left({T, \preceq_2, \tau_2}\right)$ be complete algebraic topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous  :$\forall x \in S: f \left({x}\right) = \sup \left\{ {f \left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is compact$\left.{}\right\}$
Sum of Pandigital Triplet of 3-Digit Primes
The smallest integer which is the sum of a set of $3$ three-digit primes using all $9$ digits from $1$ to $9$ once each is $999$: :$149 + 263 + 587 = 999$
Reciprocal of 1089
:$\dfrac 1 {1089} = 0 \cdotp \dot 00091 \, 82736 \, 45546 \, 37281 \, 9 \dot 1$
Square Numbers which are Sum of Sequence of Odd Cubes
The sequence of square numbers which can be expressed as the sum of a sequence of odd cubes from $1$ begins: :$1, 1225, 1 \, 413 \, 721, 1 \, 631 \, 432 \, 881, \dotsc$ {{OEIS|A046177}} The sequence of square roots of this sequence is: :$1, 35, 1189, 40 \, 391, \dotsc$ {{OEIS|A046176}}
Smallest Triplet of Consecutive Integers Divisible by Cube
The smallest sequence of triplets of consecutive integers each of which is divisible by a cube greater than $1$ is: :$\tuple {1375, 1376, 1377}$
Sequence of Composite Mersenne Numbers
The sequence of Mersenne numbers which are composite begins: :$2047, 8 \, 388 \, 607, 536 \, 870 \, 911, 137 \, 438 \, 953 \, 471, 2 \, 199 \, 023 \, 255 \, 551,\ldots$ {{OEIS|A065341}} The sequence of corresponding indices $p$ such that $2^p - 1$ is composite begins: :$11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, \ldots$ {{OEIS|A054723}} The sequence of corresponding integers $n$ such that the $n$th prime number $p \left({n}\right)$ is such that $2^{p \left({n}\right)} - 1$ is composite begins: :$5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, \ldots$ {{OEIS|A135980}}
Numbers Reversed when Multiplying by 4
Numbers of the form $\sqbrk {21 \paren 9 78}_{10}$ are reversed when they are multiplied by $4$: {{begin-eqn}} {{eqn | l = 2178 \times 4       | r = 8712 }} {{eqn | l = 21 \, 978 \times 4       | r = 87 \, 912 }} {{eqn | l = 219 \, 978 \times 4       | r = 879 \, 912 }} {{end-eqn}} and so on.
Relational Structure admits Lower Topology
Let $R = \left({S, \preceq}\right)$ be a relational structure. Then there exists a relational structure with lower topology $T = \left({S, \preceq, \tau}\right)$ such that $T$ is a topological space.
2520 equals Sum of 4 Divisors in 6 Ways
The number $2520$ can be expressed as the sum of $4$ of its divisors in $6$ different ways: {{begin-eqn}} {{eqn | l = 2520       | r = 1260 + 630 + 504 + 126 }} {{eqn | r = 1260 + 630 + 421 + 210 }} {{eqn | r = 1260 + 840 + 360 + 60 }} {{eqn | r = 1260 + 840 + 315 + 105 }} {{eqn | r = 1260 + 840 + 280 + 140 }} {{eqn | r = 1260 + 840 + 252 + 168 }} {{end-eqn}} This is the maximum possible number of ways it is possible to express an integer as the sum of $4$ of its divisors.
Operating on Ordered Group Inequalities
If $x \prec y$ and $z \prec w$, then $x \circ z \prec y \circ w$. If $x \prec y$ and $z \preceq w$, then $x \circ z \prec y \circ w$. If $x \preceq y$ and $z \prec w$, then $x \circ z \prec y \circ w$. If $x \preceq y$ and $z \preceq w$, then $x \circ z \preceq y \circ w$.
Reflexive Closure of Antisymmetric Relation is Antisymmetric
Let $S$ be a set. Let $\mathcal R$ be an antisymmetric relation on $S$. Let $\mathcal R^=$ be the reflexive closure of $\mathcal R$. Then $\mathcal R^=$ is also antisymmetric.
Conditional is not Left Self-Distributive/Formulation 2
While this holds: :$\vdash \paren {\paren {p \implies q} \implies r} \implies \paren {\paren {p \implies r} \implies \paren {q \implies r} }$ its converse does not: :$\not \vdash \paren {\paren {p \implies r} \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies r}$
Self-Distributive Law for Conditional/Formulation 2
:$\vdash \paren {p \implies \paren {q \implies r} } \iff \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Rule of Exportation/Formulation 2
:$\vdash \paren {\paren {p \land q} \implies r} \iff \paren {p \implies \paren {q \implies r} }$
Cube of 20 is Sum of Sequence of 4 Consecutive Cubes
:$20^3 = \displaystyle \sum_{k \mathop = 11}^{14} k^3$ That is: :$20^3 = 11^3 + 12^3 + 13^3 + 14^3$
Largest Integer not Sum of Two Abundant Numbers
The largest integer which is not the sum of $2$ abundant numbers is $20 \, 161$.
Smallest Integer using Three Words in English Description
The smallest integer which uses exactly $3$ words in its standard (British) English description is: :$21 \, 000$: '''twenty-one thousand''' counting hyphenations as separate words.
Carmichael Number with 4 Prime Factors
$41 \, 041$ is the smallest Carmichael number with $4$ prime factors: :$41 \, 041 = 7 \times 11 \times 13 \times 41$
Tableau Extension Lemma/General Statement/Proof 1
Let $T$ be a finite propositional tableau. Let its hypothesis set $\mathbf H$ be finite. {{:Tableau Extension Lemma/General Statement}}
Reciprocal of 142,857
:$\dfrac 1 {142 \, 857} = 0 \cdotp \dot 00000 \, \dot 7$
Numbers whose Fourth Root equals Number of Divisors
There are $4$ positive integers whose $4$th root equals the number of its divisors: {{begin-eqn}} {{eqn | l = 1       | r = 1^4       | c =  }} {{eqn | l = 625       | r = 5^4       | c =  }} {{eqn | l = 6561       | r = 9^4       | c =  }} {{eqn | l = 4 \, 100 \, 625       | r = 45^4       | c =  }} {{end-eqn}} {{OEIS|A143026}}
Smallest 5 Consecutive Primes in Arithmetic Sequence
The smallest $5$ consecutive primes in arithmetic sequence are: :$9 \, 843 \, 019 + 30 n$ for $n = 0, 1, 2, 3, 4$. Note that while there are many longer arithmetic sequences of far smaller primes, those primes are not consecutive.
Smallest Triplet of Primitive Pythagorean Triangles with Same Area
The smallest set of $3$ primitive Pythagorean triangles which all have the same area are: :the $4485-5852-7373$ triangle :the $3059-8580-9109$ triangle :the $1380-19 \, 019-19 \, 069$ triangle. That area is $13 \, 123 \, 110$.
Biconditional is Transitive/Formulation 1/Proof 2
:$p \iff q, q \iff r \vdash p \iff r$
Law of Identity/Formulation 2/Proof 2
: $\vdash p \implies p$
Exchange of Order of Indexed Summations/Rectangular Domain
Let $D = \closedint a b \times \closedint c d$ be the cartesian product. Let $f: D \to \mathbb A$ be a mapping Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j} = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b \map f {i, j}$
Sum over Union of Finite Sets
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ and $T$ be finite sets. Let $f: S \cup T \to \mathbb A$ be a mapping. Then we have the equality of summations over finite sets: :$\displaystyle \sum_{u \mathop \in S \mathop \cup T} \map f u = \sum_{s \mathop \in S} \map f s + \sum_{t \mathop \in T} \map f t - \sum_{v \mathop \in S \mathop \cap T} \map f v$
Rule of Idempotence/Conjunction/Formulation 1/Proof
: $p \dashv \vdash p \land p$
Exchange of Order of Summations over Finite Sets/Subset of Cartesian Product
Let $D\subset S \times T$ be a subset. Let $\pi_1 : D \to S$ and $\pi_2 : D \to T$ be the restrictions of the projections of $S\times T$. Then we have an equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in \pi_2 \left({\pi_1^{-1} \left({s}\right)}\right)} f \left({s, t}\right) = \sum_{t \mathop \in T} \sum_{s \mathop \in \pi_1 \left({\pi_2^{-1} \left({t}\right)}\right)} f \left({s, t}\right)$
Canonical Homomorphism to Polynomial Ring is Ring Monomorphism
Let $R$ be a commutative ring with unity. Let $(R[X], \iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$. Then the canonical homomorphism $\iota : R \to R[X]$ is a ring monomorphism.
Equivalence of Definitions of Consistent Proof System
{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}} Let $\LL_0$ be the language of propositional logic. Let $\mathscr P$ be a proof system for $\LL_0$.
Irreducible Component is Closed
Let $T = \left({S, \tau}\right)$ be a topological space. Let $Y$ be an irreducible component of $T$. Then $Y$ is closed in $T$.
Homotopy Characterisation of Simply Connected Sets
Let $T = \struct {S, \tau}$ be a topological space. Let $X$ be a subset of $S$. Then $X$ is simply connected  the following conditions hold: :$(1): \quad $ $X$ is path-connected. :$(2): \quad $ All paths in $X$ with the same initial points and final points are freely homotopic.
Largest Pandigital Square including Zero
The largest pandigital square (in the sense where pandigital includes the zero) is $9 \, 814 \, 072 \, 356$: :$9 \, 814 \, 072 \, 356 = 99 \, 066^2$
Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1/Proof
:$p \lor \paren {q \land r} \dashv \vdash \paren {p \lor q} \land \paren {p \lor r}$
Multiply Perfect Number of Order 6
The number defined as: :$n = 2^{36} \times 3^8 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$ ::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$ ::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$ is multiply perfect of order $6$.
Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 1/Proof
:$\left({q \land r}\right) \lor p \dashv \vdash \left({q \lor p}\right) \land \left({r \lor p}\right)$
Functional Equation for Completed Riemann Zeta Function
Let $\xi : \C \to \C$ be the completed Riemann zeta function. Let $s\in \C$ be a complex number. Then: :$\map \xi s = \map \xi {1 - s}$
Largest Integer whose Digits taken in Pairs all form Distinct Primes
The largest integer which has the property that every pair of its digits taken together is a distinct prime number is $619 \, 737 \, 131 \, 179$.
Smallest 22 Primes in Arithmetic Sequence
The smallest $22$ primes in arithmetic sequence are: :$11 \, 410 \, 337 \, 850 \, 553 + 4 \, 609 \, 098 \, 694 \, 200 n$ for $n = 0, 1, \ldots, 21$.
Smallest Cunningham Chain of the Second Kind of Length 13
The smallest Cunningham chain of the second kind of length $13$ is: :$758 \, 083 \, 947 \, 856 \, 951$, $1 \, 516 \, 167 \, 895 \, 713 \, 901$, $3 \, 032 \, 335 \, 791 \, 427 \, 801$, $6 \, 064 \, 671 \, 582 \, 855 \, 601$, $12 \, 129 \, 343 \, 165 \, 711 \, 201$, $24 \, 258 \, 686 \, 331 \, 422 \, 401$, $48 \, 517 \, 372 \, 662 \, 844 \, 801$, $97 \, 034 \, 745 \, 325 \, 689 \, 601$, $194 \, 069 \, 490 \, 651 \, 379 \, 201$, $388 \, 138 \, 981 \, 302 \, 758 \, 401$, $776 \, 277 \, 962 \, 605 \, 516 \, 801$, $1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601$, $3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201$
Repunit Expressed using Power of 10
The repunit number $R_n$ can be expressed as: :$R_n = \dfrac {10^n - 1} 9$
Repunit in Base 9 is Triangular
Let $m$ be a repunit base $9$. Then $m$ is a triangular number.
Largest nth Power which has n Digits
The largest $n$th power which has $n$ digits is $9^{21}$: :$9^{21} = 109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209$
Left-Truncatable Prime/Examples/357,686,312,646,216,567,629,137
The largest left-truncatable prime is $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$.
Smallest Integer Divisible by All Numbers from 1 to 100
The smallest positive integer which is divisible by each of the integers from $1$ to $100$ is: :$69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800$
Integer which is Multiplied by 9 when moving Last Digit to First
Let $N$ be the positive integer: :$N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$ $N$ is the smallest positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$.
Yoneda Embedding Theorem
Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $[C^{\operatorname{op}}, \mathbf{Set}]$ be the contravariant functor category. Then the Yoneda embedding $h_- : C \to [C^{\operatorname{op}}, \mathbf{Set}]$ is a fully faithful embedding.
Titanic Prime whose Digits are all Odd
The integer defined as: :$1358 \times 10^{3821} - 1$ is a titanic prime all of whose digits are odd. That is: :$1357 \paren 9_{3821}$ where $\paren a_b$ means $b$ instances of $a$ in a string.
Gigantic Palindromic Prime
The integer defined as: :$10^{11 \, 810} + 1 \, 465 \, 641 \times 10^{5902} + 1$ is a gigantic prime which is also palindromic. That is: :$1(0)_{5901}1465641(0)_{5901}1$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.
Existence of Topological Space which satisfies no Separation Axioms
There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied.
Relation Compatible with Group Operation is Reflexive or Antireflexive
Let $\struct {G, \circ}$ be a group. Let $\RR$ be a relation on $G$ that is compatible with $\circ$. Then $\RR$ is reflexive or antireflexive.
Existence of Compact Hausdorff Space which is not T5
There exists at least one example of a compact $T_2$ (Hausdorff) space which is not a $T_5$ space.
Existence of Hausdorff Space which is not Completely Hausdorff
There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space, but is not also a completely Hausdorff space.
Existence of Semiregular Topological Space which is not Urysohn Space
There exists at least one example of a semiregular topological space which is not a Urysohn space.
Existence of Compact Space which is not Sequentially Compact
There exists at least one example of a compact topological space which is not also a sequentially compact space.
Norm of Vector Cross Product
Let $\mathbf a$ and $\mathbf b$ be vectors in the Euclidean space $\R^3$. Let $\times$ denote the vector cross product. Then: :$(1): \quad$ $\left\Vert{ \mathbf a \times \mathbf b }\right\Vert^2 = \left\Vert{\mathbf a}\right\Vert^2 \left\Vert{\mathbf b}\right\Vert^2 - \left({\mathbf a \cdot \mathbf b}\right)^2$ :$(2): \quad$ $\left\Vert{ \mathbf a \times \mathbf b }\right\Vert = \left\Vert{\mathbf a}\right\Vert \left\Vert{\mathbf b}\right\Vert \left\vert{\sin \theta}\right\vert$ where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$, or an arbitrary number if $\mathbf a$ or $\mathbf b$ is the zero vector.
Existence of Separable Space which is not Second-Countable
There exists at least one example of a separable topological space which is not also a second-countable space.
Existence of Paracompact Space which is not Compact
There exists at least one example of a paracompact topological space which is not also a compact topological space.
Uncountable Product of First-Countable Spaces is not always First-Countable
Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a first-countable space. Then it is not necessarily the case that $\struct {S, \tau}$ is also a first-countable space.
Uncountable Product of Separable Spaces is not always Separable
Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a separable space. Then it is not necessarily the case that $\struct {S, \tau}$ is also a separable space.
Product of Metacompact Spaces is not always Metacompact
Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a metacompact space. Then it is not necessarily the case that $\struct {S, \tau}$ is also metacompact space.
Finite Irreducible Space is Path-Connected
Let $T = \left({S, \tau}\right)$ be a finite irreducible topological space. Then $T$ is path-connected.
Equivalence of Definitions of Convergent of Continued Fraction
Let $F$ be a field, such as the field of real numbers. Let $n \in \N \cup \set \infty$ be an extended natural number. Let $C = \sqbrk {a_0, a_1, a_2, \dotsc}$ be a continued fraction in $F$ of length $n$. Let $k \le n$ be a natural number. {{TFAE|def = Convergent of Continued Fraction}}
Locally Connected Space is not necessarily Locally Path-Connected
Let $T = \struct {S, \tau}$ be a topological space which is locally connected. Then it is not necessarily the case that $T$ is also an locally patj-connected space.
Binary Product in Preadditive Category is Biproduct
Let $A$ be a preadditive category. Let $a_1, a_2$ be objects of $A$. Let $(a_1 \times a_2, p_1, p_2)$ be their binary product, assuming it exists. Let $i_1 : a_1 \to a_1 \times a_2$ be the unique morphism with: :$p_1 \circ i_1 = 1 : a_1 \to a_1$ :$p_2 \circ i_1 = 0 : a_1 \to a_2$ Let $i_2 : a_1 \to a_1 \times a_2$ be the unique morphism with: :$p_1 \circ i_2 = 0 : a_2 \to a_1$ :$p_2 \circ i_2 = 1 : a_2 \to a_2$ where $1$ is the identity morphism and $0$ is the zero morphism. Then $(a_1 \times a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$.