hcju/proofwikips huggingface dataset:
Limit Points of Indiscrete Space
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space consisting of at least two points. Let $H$ be a subset of $T$ such that $H \ne \O$. Then every point of $T$ is a limit point of $H$.
Sequence in Indiscrete Space converges to Every Point
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $\sequence {s_n}$ be a sequence in $T$. Then $\sequence {s_n}$ converges to every point of $S$.
Empty Set is Nowhere Dense
Let $T = \struct {S, \tau}$ be a topological space. Then the empty set $\O$ is nowhere dense in $T$.
Interior of Open Set
Let $T = \left({S, \tau}\right)$ be a topological space. Let $U \subseteq T$ be open in $T$. Then: : $U^\circ = U \iff U \in \tau$ where $U^\circ$ is the interior of $U$. That is, a subset of $S$ is open in $T$  it equals its interior.
Interior of Subset of Indiscrete Space
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subset S$ (that is, let $H$ be a proper subset of $T$). Then: :$H^\circ = H^{\circ -} = H^{\circ - \circ} = \O$ where: :$H^\circ$ denotes the interior of $H$ :$H^-$ denotes the closure of $H$.
Closure of Subset of Indiscrete Space
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $\O \subsetneq H \subseteq S$ (that is, let $H$ be a non-empty subset of $T$). Then: :$H^- = H^{- \circ} = H^{- \circ -} = S$ where: :$H^\circ$ denotes the interior of $H$ :$H^-$ denotes the closure of $H$.
Boundary of Subset of Indiscrete Space
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $\O \subsetneq H \subsetneq S$ (that is, let $H$ be a non-empty proper subset of $T$). Then: :$\partial H = S$ where $\partial H$ denotes the boundary of $H$. If $H = \O$ or $H = S$ then $\partial H = \O$.
Boundary of Boundary of Subset of Indiscrete Space
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$. Then: :$\map \partial {\partial H} = \O$ where $\partial H$ denotes the boundary of $H$.
Indiscrete Space is Separable
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space such that $S$ has more than one element. Then $T$ is separable.
Mapping to Indiscrete Space is Continuous
Let $T_1 = \struct {S_1, \tau_1}$ be any topological space. Let $T_2 = \struct {S_2, \tau_2}$ be the indiscrete topological space on $S_2$. Let $\phi: S_1 \to S_2$ be a mapping. Then $\phi$ is continuous.
Indiscrete Space is Path-Connected
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is path-connected.
Indiscrete Space is Connected
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is connected.
Indiscrete Space is Arc-Connected iff Uncountable
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is arc-connected  $S$ is an uncountable set.
Indiscrete Space is Irreducible
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is irreducible.
Indiscrete Space is Ultraconnected
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is ultraconnected.
Indiscrete Non-Singleton Space is not T0
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space which has more than one element. Then $T$ is not a $T_0$ (Kolmogorov) space.
Indiscrete Space is T5
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is a $T_5$ space.
Indiscrete Space is T3
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space . Then $T$ is a $T_3$ space.
Indiscrete Space is Pseudometrizable
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is pseudometrizable.
Partition Topology is Topology
Let $S$ be a set. Let $\PP$ be a partition of $S$. Let $\tau$ be the set of subsets of $S$ defined as: :$a \in \tau \iff a$ is the union of sets of $\PP$ Then $\tau$ is a topology on $S$.
Basis for Partition Topology
Let $S$ be a set. Let $\PP$ be a partition of $S$. Let $\tau$ be the partition topology on $S$ defined as: :$a \in \tau \iff a$ is the union of sets of $\mathcal P$ Then $\PP$ forms a basis of $\tau$.
Open Set in Partition Topology is also Closed
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a partition space : :$\forall U \subseteq S: U \in \tau \iff \relcomp S U \in \tau$ That is, a topological space is a partition space  all open sets are closed, and all closed sets are also open.
Quotient Topology of Partition Topology is Discrete Space
Let $\PP$ be a partition of a set $S$. Let $T = \struct {S, \tau}$ be the partition space formed from $\PP$. Let $S / \PP$ be the quotient set of $S$ by $\PP$. Then the quotient topology $\tau_{S / \PP}$ is a discrete topology.
Partition of Singletons yields Discrete Topology
Let $S$ be a set which is non-empty. Let $\PP$ be the (trivial) partition of singletons on $S$: :$\PP = \set {\set x: x \in S}$ Then the partition topology on $\PP$ is the discrete topology.
Partition Topology is not T0
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_0$ (Kolmogorov) space.
Partition Topology is T5
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is a $T_5$ space.
Partition Topology is T3 1/2
Let $S$ be a set and let $\PP$ be a partition on $S$. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is a $T_{3 \frac 1 2}$ space.
Odd-Even Topology is Second-Countable
Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$. Then $T$ is second-countable.
Odd-Even Topology is Weakly Countably Compact
Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$. Then $T$ is weakly countably compact.
Odd-Even Topology is not Countably Compact
Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$. Then $T$ is not countably compact.
Countable Discrete Space is not Weakly Countably Compact
Let $T = \struct {S, \tau}$ be a countable discrete space. Then $T$ is not weakly countably compact.
Weak Countable Compactness is not Preserved under Continuous Maps
Let $T_A = \struct {S_A, \tau_A}$ be a topological space which is weakly countably compact. Let $T_B = \struct {S_B, \tau_B}$ be another topological space. Let $\phi: T_A \to T_B$ be a continuous mapping. Then $T_B$ is not necessarily weakly countably compact.
Deleted Integer Topology is not Countably Compact
Let $S = \R_{\ge 0} \setminus \Z$, and let $\tau$ be the deleted integer topology on $S$. Then the topological space $T = \struct {S, \tau}$ is not countably compact.
Deleted Integer Topology is Second-Countable
Let $S = \R_{\ge 0} \setminus \Z$. Let $\tau$ be the deleted integer topology on $S$. Then the topological space $T = \struct {S, \tau}$ is second-countable.
Deleted Integer Topology is Weakly Countably Compact
Let $S = \R_{\ge 0} \setminus \Z$. Let $\tau$ be the deleted integer topology on $S$. Then the topological space $T = \struct {S, \tau}$ is weakly countably compact.
Pseudometric induces Topology
Let $S \ne \O$ be a non-empty set. Consider a pseudometric space $\struct {S, d}$ where $d: S \times S \to \R_{\ge 0}$ is a pseudometric. Then $\struct {S, d}$ gives rise to a topological space $\struct {S, \tau_d}$ whose topology $\tau_d$ is '''defined''' (or '''induced''') by $d$.
Open Sets in Pseudometric Space
Let $P = \left({A, d}\right)$ be a pseudometric space. Then $\varnothing$ and $A$ are both open in $P$.
Partition Space is Pseudometrizable
Let $T = \struct {S, \tau}$ be a partition space. Then $T$ is pseudometrizable.
Double Pointed Topology is not T0
Let $T_1 = \struct {S, \tau_S}$ be a topological space. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$. Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$. Then $T$ is not a $T_0$ (Kolmogorov) space.
Double Pointed Discrete Real Number Space is Weakly Countably Compact
Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers. 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 the double pointed (uncountable) discrete space which is the product space of $T_\R$ and $T_D$. Then $T$ is weakly countably compact.
Double Pointed Discrete Real Number Space is not Lindelöf
Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers. 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 the double pointed (uncountable) discrete space which is the product space of $T_\R$ and $T_D$. Then $T$ is '''not''' a Lindelöf space.
Particular Point Topology is Topology
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $\tau_p$ is a topology on $S$, and $T$ is a topological space.
Limit Points in Particular Point Space
Let $T = \struct {S, \tau_p}$ be a particular point space. Let $x \in S$ such that $x \ne p$. Then $x$ is a limit point of $p$.
Closure of Open Set of Particular Point Space
Let $T = \struct {S, \tau_p}$ be a particular point space. Let $U \in \tau_p$ be open in $T$ such that $U \ne \O$. Then: :$U^- = S$ where $U^-$ denotes the closure of $U$.
Interior of Closed Set of Particular Point Space
Let $T = \struct {S, \tau_p}$ be a particular point space. Let $V \subseteq S$ be closed in $T$ such that $V \ne S$. Then: :$V^\circ = \O$ where $V^\circ$ denotes the interior of $V$.
Point in Particular Point Space is not Omega-Accumulation Point
Let $T = \struct {S, \tau_p}$ be a particular point space. Let $x \in S$ such that $x \ne p$. Let $H \subseteq S$ such that $p \in H$. Then $x$ is not an $\omega$-accumulation point of $H$.
Particular Point Space is T0
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is a $T_0$ (Kolmogorov) space.
Non-Trivial Particular Point Topology is not T1
Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton. Then $T$ is not a $T_1$ (Fréchet) space.
Particular Point Topology with Three Points is not T4
Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton or a doubleton. That is, such that $S$ has more than two distinct elements. Then $T$ is not a $T_4$ space.
Compact Space in Particular Point Space
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $\set p$ is compact in $T$.
Finite Multiplicative Subgroup of Field is Cyclic
Let $\left({F, +, \times}\right)$ be a field. Let $\left({F^*, \times}\right)$ denote the multiplicative group of $F$. Let $C$ be a finite subgroup of $\left({F^*, \times}\right)$. Then $C$ is cyclic.
Induced Homomorphism of Polynomial Forms
Let $R$ and $S$ be commutative rings with unity. Let $\phi : R \to S$ be a ring homomorphism. Let $R \left[{X}\right]$ and $S \left[{X}\right]$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$. Then the map $\overline{\phi}: R \left[{X}\right] \to S \left[{X}\right]$ given by: :$\overline{\phi} \left({a_0 + a_1 X + \cdots + a_n X^n}\right) = \phi \left({a_0}\right) + \phi \left({a_1}\right) X + \cdots + \phi \left({a_n}\right) X^n$ is a ring homomorphism.
Structure of Simple Algebraic Field Extension
Let $F / K$ be a field extension. Let $\alpha \in F$ be algebraic over $K$. Let $\mu_\alpha$ be the minimal polynomial of $\alpha$ over $K$. Let $K \left[{\alpha}\right]$ (resp. $K \left({\alpha}\right)$) be the subring (resp. subfield) of $F$ generated by $K \cup \left\{ {\alpha}\right\}$. Then: :$K \left[{\alpha}\right] = K \left({\alpha}\right) \simeq K \left[{X}\right] / \left\langle{\mu_\alpha}\right\rangle$ where $\left\langle{\mu_\alpha}\right\rangle$ is the ideal of the ring of polynomial functions generated by $\mu_\alpha$. {{explain|Link to Generator needs refining}} Moreover: :$n := \left[{K \left({\alpha}\right) : K}\right] = \deg \mu_\alpha$ and: :$1, \alpha, \dotsc, \alpha^{n - 1}$ is a basis of $K \left({\alpha}\right)$ over $K$.
Subset of Particular Point Space is either Open or Closed
Let $T = \left({S, \tau_p}\right)$ be a particular point space. Let $H \subseteq S$ be any subset of $T$. Then $H$ is either open or closed in $T$. The only sets which are both closed and open in $T$ are $S$ and $\varnothing$.
Particular Point Space is Weakly Locally Compact
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is weakly locally compact.
Infinite Particular Point Space is not Compact
Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not compact.
Uncountable Particular Point Space is not Lindelöf
Let $T = \struct {S, \tau_p}$ be an uncountable particular point space. Then $T$ is not a Lindelöf space.
Particular Point Space is Separable
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is separable.
Field Adjoined Set
Let $F$ be a field. Let $S \subseteq F$ be a subset of $F$. Let $K \le F$ be a subfield of $F$. The subring $K \sqbrk S$ of $F$ generated by $K \cup S$ is the set of all finite linear combinations of powers of elements of $S$ with coefficients in $K$. The subfield $\map K S$ of $F$ generated by $K \cup S$ is the set of all $x y^{-1} \in F$ with $a, b \in K \sqbrk S$, $b \ne 0$. $\map K S$ is isomorphic to the field of quotients $Q$ of $K \sqbrk S$.
Separability in Uncountable Particular Point Space
Let $T = \struct {S, \tau_p}$ be an uncountable particular point space. Let $H = S \setminus \set p$ where $\setminus$ denotes set difference. Then $H$ is not separable.
Uncountable Particular Point Space is not Second-Countable
Let $T = \struct {S, \tau_p}$ be an uncountable particular point space. Then $T$ is not second-countable.
Particular Point Space is First-Countable
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is first-countable.
Homeomorphic Non-Comparable Particular Point Topologies
Let $S$ be a set with at least two elements. Let $p, q \in S: p \ne q$. Let $\tau_p$ and $\tau_q$ be the particular point topologies on $S$ by $p$ and $q$ respectively. Then the topological spaces $T_p = \struct {S, \tau_p}$ and $T_q = \struct {S, \tau_q}$ are homeomorphic. However, $\tau_p$ and $\tau_q$ are not comparable.
Universal Property for Simple Field Extensions
Let $F / K$ be a field extension. Let $\alpha \in F$ be an algebraic over $K$. Let $\mu_\alpha$ be the minimal polynomial over $\alpha$ over $K$. Let $\psi : K \left({\alpha}\right) \to L$ be a homomorphism. Let $\phi = \psi \restriction_K$. Let $\overline \phi: K \left[{X}\right] \to L \left[{X}\right]$ be the Induced Homomorphism of Polynomial Forms. Then $\psi\left({\alpha}\right)$ is a root of $\overline \phi \left({\mu_\alpha}\right)$ in $L$. Conversely let $L$ be a field. Let $\phi: K \to L$ be a homomorphism. Let $\beta$ be a root of $\overline \phi  \left({\mu_\alpha}\right)$ in $L$. Then there exists a unique field homomorphism $\psi : K \left({\alpha}\right) \to L$ extending $\phi$ that sends $\alpha$ to $\beta$.
Finite Field Extension is Algebraic
Let $L / K$ be a finite field extension. Then $L / K$ is algebraic.
Finitely Generated Algebraic Extension is Finite
Let $L/K$ be a field extension and $\alpha_1,\ldots,\alpha_n \in L$ algebraic over $K$. Then $K \left({\alpha_1, \ldots, \alpha_n}\right) / K$ is a finite field extension.
Maximal Algebraic Extension is Subfield
Let $L/K$ be a extension of fields. Let $K^a$ be the maximal algebraic extension of $K$ contained in $L$. Then $K^a$ is a subfield of $L$.
Closed Set in Particular Point Space has no Limit Points
Let $T = \struct {S, \tau_p}$ be a particular point space. Let $H \subsetneq S$ be closed in $T$. Then $H$ has no limit points.
Particular Point Space is Scattered
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is scattered.
Particular Point Space is Irreducible
Let $T = \left({S, \tau_p}\right)$ be a particular point space. Then $T$ is irreducible.
Particular Point Space is not Ultraconnected
Let $T = \struct {S, \tau_p}$ be a particular point space with at least three points. Then $T$ is not ultraconnected.
Dispersion Point in Particular Point Space
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $p$ is dispersion point of $T$.
Infinite Particular Point Space is not Weakly Countably Compact
Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not weakly countably compact.
Fundamental Theorem of Symmetric Polynomials
Let $K$ be a field. Let $f$ a symmetric polynomial over $K$. Then $f$ can be written uniquely as a polynomial in the elementary symmetric polynomials.
Particular Point Space is Pseudocompact
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is pseudocompact.
Particular Point Space is Path-Connected
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is path-connected.
Particular Point Space is not Arc-Connected
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is not arc-connected.
Basis for Particular Point Space
Let $T = \left({S, \tau_p}\right)$ be a particular point space. Consider the set $\mathcal B$ defined as: :$\mathcal B = \left\{{\left\{{x, p}\right\}: x \in S}\right\} \cup \left\{{p}\right\}$ Then $B$ is a basis for $S$.
Particular Point Space is Non-Meager
Let $T = \left({S, \tau_p}\right)$ be a particular point space. Then $T$ is non-meager.
Infinite Particular Point Space is not Countably Metacompact
Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not countably metacompact.
Exportation and Self-Conditional
:$p \land q \implies r \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$
Łoś-Vaught Test
Let $T$ be a satisfiable $\mathcal L$-theory with no finite models. Let $T$ be $\kappa$-categorical for some infinite cardinal $\kappa \ge \left|{\mathcal L}\right|$. Then $T$ is complete.
Sierpiński Space is Irreducible
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is irreducible.
Sierpiński Space is Path-Connected
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is path-connected.
Sierpiński Space is not Arc-Connected
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is not arc-connected.
Sierpiński Space is T5
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is a $T_5$ space.
Sierpiński Space is T4
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is a $T_4$ space.
Upward Löwenheim-Skolem Theorem
{{Disambiguate|Definition:Model|I suspect model of a first-order theory $\LL$, which is more specific than what is linked to now}} Let $T$ be an $\LL$-theory with an infinite model. Then for each infinite cardinal $\kappa \ge \card \LL$, there exists a model of $T$ with cardinality $\kappa$.
Compactness Theorem
Let $\LL$ be the language of predicate logic. Let $T$ be a set of $\LL$-sentences. Then $T$ is satisfiable  $T$ is finitely satisfiable.
Łoś's Theorem
Let $\LL$ be a language. Let $I$ be an infinite set. Let $\UU$ be an ultrafilter on $I$. Let $\map \phi {v_1, \ldots, v_n}$ be an $\LL$-formula. Let $\MM$ be the ultraproduct: :$\displaystyle \paren {\prod_{i \mathop \in I} \MM_i} / \UU$ where each $\MM_i$ is an $\LL$-structure. Then, for all $m_1 = \paren {m_{1, i} }_\UU, \dots, m_n = \paren {m_{n, i} }_\UU$ in $\MM$: :$\MM \models \map \phi {m_1, \ldots, m_n}$ : :the set $\set {i \in I: \MM_i \models \map \phi {m_{1, i}, \ldots, m_{n, i} } }$ is in $\UU$. In particular, for all $\LL$-sentences $\phi$, we have that: :$\MM \models \phi$  $\set {i \in I: \MM_i \models \phi}$ is in $\UU$.
Closed Extension Topology is Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $\tau^*_p$ be the closed extension topology of $\tau$. Then $\tau^*_p$ is a topology on $S^*_p = S \cup \set p$.
Quantifier Free Formula is Preserved by Superstructure
Let $\MM, \NN$ be $\LL$-structures such that  $\MM$ is a substructure of $\NN$. {{Disambiguate|Definition:Structure}} Let $\map \phi {\bar x}$ be a quantifier-free $\LL$-formula, and let $\bar a \in\MM$. Then $\MM \models \map \phi {\bar a}$  $\NN \models \map \phi {\bar a}$.
Algebraically Closed Field is Infinite
Let $F$ be an algebraically closed field. Then $F$ is infinite.
Field of Uncountable Cardinality K has Transcendence Degree K
Let $F$ be a field of uncountable cardinality $\kappa$. Then $F$ has transcendence degree $\kappa$ over its prime field.
Theory of Algebraically Closed Fields of Characteristic p is Complete
Let $p$ be either $0$ or a prime number. Let $ACF_p$ be the theory of algebraically closed fields of characteristic $p$ in the language $\mathcal L_r = \left\{ {0, 1, +, -, \cdot}\right\}$ for rings, where: :$0, 1$ are constants and: :$+, -, \cdot$ are binary functions.  Then $ACF_p$ is complete.
Lefschetz Principle (First-Order)
Let $\phi$ be a sentence in the language $\LL_r = \set {0, 1, +, -, \cdot}$ for rings, where $0, 1$ are constants and $+, -, \cdot$ are binary functions.  :$(1): \quad \phi$ is true in every algebraically closed field of characteristic $0$. :$(2): \quad \phi$ is true in some algebraically closed field of characteristic $0$. :$(3): \quad \phi$ is true in algebraically closed fields of characteristic $p$ for arbitrarily large primes $p$. :$(4): \quad \phi$ is true in algebraically closed fields of characteristic $p$ for sufficiently large primes $p$. Note in particular that since $\C$ is an algebraically closed field of characteristic $0$, these are equivalent to $\phi$ being true in the field $\C$.
CNF Satisfiability Problem is NP-Complete
The conjunctive normal form boolean satisfiability problem (CNF SAT) is NP-complete.
Closed Sets of Closed Extension Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then the closed sets of $T^*_p$ (apart from $S^*_p$) are the closed sets of $T$. This explains why $\tau^*_p$ is called the closed extension topology of $\tau$.
Particular Point Topology is Closed Extension Topology of Discrete Topology
Let $S$ be a set and let $p \in S$. Let $\tau_p$ be the particular point topology on $S$. Let $T = \struct {S \setminus \set p, \vartheta}$ be the discrete topological space on $S \setminus \set p$. Then $T^* = \struct {S, \tau_p}$ is a closed extension space of $T$.
Closed Extension Topology is not T1
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_1$ (Fréchet) space.
Algebraic Numbers are Countable
The set $\Bbb A$ of algebraic numbers is countable.
Limit Points in Closed Extension Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Let $x \in S$. Then $x$ is a limit point of $p$.
Closure of Open Set of Closed Extension Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $U^- = S^*_p$ where $U^-$ denotes the closure of $U$ in $T^*_p$.
Closed Extension Space is Irreducible
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is irreducible.
Polynomial over Field has Finitely Many Roots
Let $F$ be a field. Let $F \left[{x}\right]$ be the ring of polynomial functions in the indeterminate $x$. If $p \in F \left[{x}\right]$ be non-null, then $p$ has finitely many roots in $F$.
Set of Polynomials over Infinite Set has Same Cardinality
Let $S$ be a set of infinite cardinality $\kappa$. Let $S \left[{x}\right]$ be the set of polynomial forms over $S$ in the indeterminate $x$. Then $S \left[{x}\right]$ has cardinality $\kappa$.