hcju/proofwikips huggingface dataset:
Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods
Let $T = \struct {S, \tau}$ be a Hausdorff space. Let $V_1$ and $V_2$ be compact sets in $T$. Then $V_1$ and $V_2$ have disjoint neighborhoods.
Compact Subsets of T3 Spaces
Let $T = \struct {S, \tau}$ be a $T_3$ space. Let $A \subseteq S$ be compact in $T$. Then for each $U \in \tau$ such that $A \subseteq U$: :$\exists V \in \tau: A \subseteq V \subseteq V^- \subseteq U$ where $V^-$ denotes the closure of $V$.
Compact Hausdorff Space is T4
Let $T = \struct {S, \tau}$ be a compact Hausdorff space. Then $T$ is a $T_4$ space.
Maximum Cardinality of Separable Hausdorff Space
Let $T = \struct {S, \tau}$ be a Hausdorff space which is separable. Then $S$ can have a cardinality no greater than $2^{2^{\aleph_0} }$.
Cluster Point of Ultrafilter is Unique
Let $S$ be a set. Let $\FF$ be an ultrafilter on $S$. Let $x \in S$ be a cluster point of $\FF$. Then there is no point $y \in S: y \ne x$ such that $y$ is also a cluster point of $\FF$.
Sequence of Implications of Separation Axioms
Let $P_1$ and $P_2$ be separation axioms and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies separation axiom $P_1$, then $T$ also satisfies separation axiom $P_2$. Then the following sequence of separation axioms holds: {| |- | align="center" | || | align="center" | || | align="center" | Perfectly Normal || | align="center" | $\implies$ || | align="center" | Perfectly $T_4$ || | align="center" | $\implies$ || | align="center" | $T_4$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | || | align="center" | || | align="center" | Completely Normal || | align="center" | $\implies$ || | align="center" | $T_5$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | Normal || | align="center" | $\implies$ || | align="center" | $T_4$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | $T_{3 \frac 1 2}$ || | align="center" | $\impliedby$ || | align="center" | Completely Regular (Tychonoff) || | align="center" | $\implies$ || | align="center" | Urysohn || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | $T_3$ || | align="center" | $\impliedby$ || | align="center" | Regular || | align="center" | $\implies$ || | align="center" | $T_{2 \frac 1 2}$ (Completely Hausdorff) || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | Semiregular || | align="center" | $\implies$ || | align="center" | $T_2$ (Hausdorff) || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $T_1$ (Fréchet) || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $T_0$ (Kolmogorov) || |}
Sequence of Implications of Global Compactness Properties
Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | || | align="center" | || | align="center" | Sequentially Compact || | align="center" | || | align="center" | || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | Compact || | align="center" | $\implies$ || | align="center" | Countably Compact || | align="center" | $\implies$ || | align="center" | Pseudocompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | $\sigma$-Compact || | align="center" | || | align="center" | Weakly Countably Compact || | align="center" | || | align="center" | || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |- | align="center" | Lindelöf Space || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |}
Sequence of Implications of Local Compactness Properties
Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Compact || | align="center" | $\implies$ || | align="center" | Strongly Locally Compact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Weakly $\sigma$-Locally Compact || | align="center" | $\implies$ || | align="center" | Weakly Locally Compact || | align="center" | $\Longleftarrow$ || | align="center" | Locally Compact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | $\sigma$-Compact || | align="center" | || | align="center" | || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | Lindelöf Space || | align="center" | || | align="center" | || |}
Equivalence of Definitions of Integral Dependence
Let $A$ be an extension of a commutative ring with unity $R$. For $x \in A$, the following are equivalent: {{begin-eqn}} {{eqn | n = 1       | o =       | r = \) $x$ is integral over $R$ \( }} {{eqn | n = 2       | o =       | r = \) The $R$-module $R \left[{x}\right]$ is finitely generated \( }} {{eqn | n = 3       | o =       | r = \) There exists a subring $B$ of $A$ such that $x \in B$, $R \subseteq B$ and $B$ is a finitely generated $R$-module \( }} {{eqn | n = 4       | o =       | r = \) There exists a subring $B$ of $A$ such that $x B \subseteq B$ and $B$ is finitely generated over $R$ \( }} {{eqn | n = 5       | o =       | r = \) There exists a faithful $R \left[{x}\right]$-module $B$ that is finitely generated as an $R$-module \( }} {{end-eqn}}
Transitivity of Finite Generation
Let $A \subseteq B \subseteq C$ be rings. Suppose $B$ is a finitely generated $A$-module, and $C$ is a finitely generated $B$-module. Then $C$ is a finitely generated $A$-module.
Integral Closure is Subring
Let $A$ be an extension of a commutative ring with unity $\struct {R, +, \circ}$. Let $C$ be the integral closure of $R$ in $A$. Then $C$ is a subring of $A$.
Localization of Ring is Unique
Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset. Let $\left({A_S, \iota}\right)$ and $\left({\tilde A_S, \tilde \iota}\right)$ both satisfy the definition of the localization of $A$ at $S$. Then there is a canonical isomorphism $\phi: A_S \to \tilde A_S$.
Spectrum of Ring is Nonempty
Let $A$ be a non-trivial commutative ring with unity. Then its prime spectrum is non-empty: :$\Spec A \ne \O$
Compact Space is Strongly Locally Compact
Let $T = \struct {S, \tau}$ be a compact space. Then $T$ is a strongly locally compact space.
Compact Space is Weakly Sigma-Locally Compact
Let $T = \struct {S, \tau}$ be a compact space. Then $T$ is a weakly $\sigma$-locally compact space.
Sequence of Implications of Paracompactness Properties
Let $P_1$ and $P_2$ be paracompactness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | || | align="center" | || | align="center" | Compact || | align="center" | $\implies$ || | align="center" | Countably Compact || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Fully Normal || | align="center" | $\implies$ || | align="center" | Paracompact || | align="center" | $\implies$ || | align="center" | Countably Paracompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Fully $T_4$ || | align="center" | || | align="center" | Metacompact || | align="center" | $\implies$ || | align="center" | Countably Metacompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |- | align="center" | $T_4$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |}
Compactness Properties in Hausdorff Spaces
Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a $T_2$ (Hausdorff) space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Fully $T_4$ || | align="center" | $\iff$ || | align="center" | Paracompact || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Weakly $\sigma$-Locally Compact || | align="center" | $\implies$ || | align="center" | $T_4$ || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Weakly Locally Compact || | align="center" | $\implies$ || | align="center" | $T_{3 \frac 1 2}$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | First-Countable and Countably Compact|| | align="center" | $\implies$ || | align="center" | $T_3$ || |}
Zero Locus of Set is Zero Locus of Generated Ideal
Let $k$ be a field. Let $n\geq1$ be a natural number. Let $A = k \left[{X_1, \ldots, X_n}\right]$ be the ring of polynomial functions in $n$ variables over $k$. Let $T \subseteq A$ be a subset, and $V \left({T}\right)$ the zero locus of $T$. Let $J = \left({T}\right)$ be the ideal generated by $T$. Then: : $V \left({T}\right) = V \left({J}\right)$
Compactness Properties in T3 Spaces
Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a $T_3$ space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Second-Countable || | align="center" | $\implies$ || | align="center" | Lindelöf || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | $\Downarrow$|| | align="center" | || | align="center" | Fully $T_4$ $\iff$ Paracompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | $T_5$ || | align="center" | $\implies$ || | align="center" | $T_4$ || |- |}
Finite Product of Sigma-Compact Spaces is Sigma-Compact
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a finite set of topological spaces. Let $\displaystyle \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_i}$ be the product space of $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$. Let each of $\struct {S_i, \tau_i}$ be $\sigma$-compact. Then $\struct {S, \tau}$ is also $\sigma$-compact.
Countable Product of Sequentially Compact Spaces is Sequentially Compact
Let $I$ be an indexing set with countable 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 sequentially compact. Then $\struct {S, \tau}$ is also sequentially compact.
Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact
Let $n \in \Z_{\ge 0}$ be a (strictly) positive integer. Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a finite set of topological spaces. Let $\displaystyle \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_i}$ be the product space of $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$. Let each of $\struct {S_i, \tau_i}$ be weakly locally compact. Then $\struct {S, \tau}$ is also weakly locally compact.
Countable Product of Second-Countable Spaces is Second-Countable
Let $I$ be an indexing set with countable 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 second-countable. Then $\struct {S, \tau}$ is also second-countable.
Countable Product of Separable Spaces is Separable
Let $I$ be an indexing set with countable 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 separable. Then $\struct {S, \tau}$ is also separable.
Compactness Properties Preserved under Continuous Surjection
Let $T_A = \strict {S_A, \tau_A}$ and $T_B = \strict {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ has one of the following properties, then $T_B$ has the same property: :Compact Space :$\sigma$-Compact Space :Countable Compact Space :Sequential Compact Space :Lindelöf Space
Projection from Product Topology is Open
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the Tychonoff topology on $S$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Then both $\pr_1$ and $\pr_2$ are open.
Weak Local Compactness is Preserved under Open Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous mapping which is also an open mapping and a surjection. If $T_A$ is weakly locally compact, then $T_B$ is also weakly locally compact.
Compactness Properties Preserved under Projection Mapping
Let $I$ be an indexing set with countable 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 $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate. Then $\pr_\alpha$ preserves the following compactness properties. That is, if $\struct {S, \tau}$ has one of the following properties, then each of $\struct {S_\alpha, \tau_\alpha}$ has the same property: :Compact Space :$\sigma$-Compact Space :Countably Compact Space :Sequentially Compact Space :Lindelöf Space :Locally Compact Space :Weakly Locally Compact Space :Paracompact Space
Countability Properties Preserved under Projection Mapping
Let $\sequence{\struct{S_\alpha, \tau_\alpha}}$ be a sequence of topological spaces. Let $\displaystyle \struct{S, \tau} = \prod \struct{S_\alpha, \tau_\alpha}$ be the product space of $\sequence {\struct{S_\alpha, \tau_\alpha}}$. Let $\pr_\alpha: \struct{S, \tau} \to \struct{S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate. Then $\pr_\alpha$ preserves the following countability properties. That is, if $\struct{S, \tau}$ has one of the following properties, then each of $\struct{S_\alpha, \tau_\alpha}$ has the same property. :Separability :First-Countability :Second-Countability
Countability Axioms Preserved under Open Continuous Surjection
Let $T_A = \left({S_A, \tau_A}\right)$ and $T_B = \left({S_B, \tau_B}\right)$ be topological spaces. Let $\phi: T_A \to T_B$ be a surjective open mapping which is also continuous. If $T_A$ has one of the following properties, then $T_B$ has the same property: :First-Countability :Second-Countability
Connectedness of Points is Equivalence Relation
Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff a$ is connected to $b$ where $a, b \in S$. Then $\sim$ is an equivalence relation.
Surgery for Rings
Let $R$ and $S$ be commutative rings with unity, and $\phi: R \to S$ a ring monomorphism. Then there is a ring $T$ isomorphic to $S$ that contains $R$ as a subring.
Clopen Set contains Components of All its Points
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be both closed and open in $T$. Then $H$ contains the components of all of its points.
Connectedness Between Two Points is an Equivalence Relation
Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff \exists S \subseteq T: S$ is connected between the two points  $a$ and $b$ where $a, b \in X$. Then $\sim$ is an equivalence relation.
Quasicomponent is Intersection of Clopen Sets
Let $T = \struct {S, \tau}$ be a topological space. Let $p \in S$. Then the quasicomponent containing $p$ equals the intersection of all sets which are both open and closed which contain $p$.
Localization of Ring Exists
Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset with $0 \notin S$. Then there exists a localization $\struct {A_S, \iota}$ of $A$ at $S$.
Transitivity of Integrality
Let $A \subseteq B \subseteq C$ be extensions of commutative rings with unity. Suppose that $C$ is integral over $B$, and $B$ is integral over $A$. Then $C$ is integral over $A$.
Universal Property for Field of Quotients
Let $\struct {D, +, \circ}$ be an integral domain. Let $\struct {F, \oplus, \cdot}$ be a field of quotients of $D$. Then $F$ satisfies the following universal property: There exists a (ring) homomorphism $\iota : D \to F$ such that: ::for every field $\tilde F$ and :and: ::for every (ring) homomorphism $\phi: D \to \tilde F$ :there exists a unique field homomorphism $\psi: F \to \tilde F$ satisfying: :::$\psi \iota = \phi$ That is, the following diagram commutes: :Universal Property Namely we may take: :$\psi: a / b \mapsto \map \phi a \map \phi b^{-1}$
Constant Mapping is Continuous
Let $T_A = \left({A, \tau_A}\right)$ and $T_B = \left({B, \tau_B}\right)$ be topological spaces. Let $b \in B$ be any point in $B$. Let $f_b: A \to B$ be the constant mapping defined by: :$\forall x \in A: f_b \left({x}\right) = b$ Then $f_b$ is continuous.
Path-Connectedness is Equivalence Relation
Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff a$ is path-connected to $b$ where $a, b \in S$. Then $\sim$ is an equivalence relation.
Arc-Connectedness is Equivalence Relation
Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff a$ is arc-connected to $b$ where $a, b \in S$. Then $\sim$ is an equivalence relation.
Joining Arcs makes Another Arc
Let $T$ be a topological space. Let $\mathbb I \subseteq \R$ be the closed unit interval $\left[{0 \,.\,.\, 1}\right]$. Let $a,b,c$ be three distinct points of $T$. Let $f, g: \mathbb I \to T$ be arcs in $T$ from $a$ to $b$ and from $b$ to $c$ respectively. Let $h: \mathbb I \to T$ be the mapping given by: :$h \left({x}\right) = \begin{cases} f \left({2 x}\right) & : x \in \left[{0 \,.\,.\, \dfrac 1 2}\right] \\ g \left({2 x - 1}\right) & : x \in \left[{\dfrac 1 2 \,.\,.\, 1}\right] \end{cases}$ Then either: :$h$ is an arc in $T$ or :There exists some restriction of $h$ which, possibly after reparametrisation, is an arc in $T$.
Arc in Topological Space is Path
Let $T = \struct {X, \tau}$ be a topological space. Let $f$ be an arc in $T$. Then $f$ is a path in $T$.
Vector Times Magnitude Same Length As Magnitude Times Vector
Let $\mathbf u$ and $\mathbf v$ be two vectors in the vector space $\struct {G, +_G, \circ}_K$ over a division ring $\struct {K, +_K, \times}$ with subfield $\R$ such that $\R \subseteq \map Z K$ with $\map Z K$ the center of $K$ Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be the lengths of $\mathbf u$ and $\mathbf v$ respectively. Then: :$\norm {\paren {\norm {\mathbf v} \circ \mathbf u} } = \norm {\paren {\norm {\mathbf u} \circ \mathbf v} }$
Increasing Sequence in Ordered Set Terminates iff Maximal Element
Let $\left({P, \le}\right)$ be an ordered set.  : $(1): \quad$ Every increasing sequence $x_1 \leq x_2 \leq x_3 \leq \cdots$ with $x_i \in P$ eventually terminates: there is $n \in \N$ such that $x_n = x_{n+1} = \cdots$. : $(2): \quad$ Every non-empty subset of $P$ has a maximal element.
Characterisation of UFDs
Let $A$ be an integral domain.  : $(1): \quad A$ is a unique factorisation domain  : $(2): \quad A$ is a GCD domain satisfying the ascending chain condition on principal ideals. : $(3): \quad A$ satisfies the ascending chain condition on principal ideals and every irreducible element of $A$ is a prime element of $A$.
Ultraconnected Space is Path-Connected
Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is path-connected.
Continuous Real-Valued Function on Irreducible Space is Constant
Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Let $f: S \to \R$ be a continuous real-valued function. Then $f$ is constant, that is: :$\exists a \in \R: \forall x \in S: \map f x = a$
Irreducible Space is Pseudocompact
Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Then $T$ is pseudocompact.
Arc-Connected Space is Path-Connected
Let $T = \struct {S, \tau}$ be a topological space which is arc-connected. Then $T$ is path-connected.
Irreducible Space is Connected
Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Then $T$ is connected.
Non-Trivial Ultraconnected Space is not T1
Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. If $S$ has more than one element, then $T$ is not a $T_1$ (Fréchet) space. That is, if $T$ is a $T_1$ (Fréchet) space with more than one element, it is not ultraconnected.
Dot Product of Vector with Itself
:$\mathbf u \cdot \mathbf u = \norm {\mathbf u}^2$
Cosine Formula for Dot Product
Let $\mathbf v,\mathbf w$ be two non-zero vectors in $\R^n$. The dot product of $\mathbf v$ and $\mathbf w$ can be calculated by: :$\mathbf v \cdot \mathbf w = \norm {\mathbf v} \norm {\mathbf w} \cos \theta$ where: :$\norm {\, \cdot \,}$ denotes vector length and :$\theta$ is the angle between $\mathbf v$ and $\mathbf w$.
Quasicomponents and Components are Equal in Locally Connected Space
Let $T = \struct {S, \tau}$ be a topological space which is locally connected. Then $A \subseteq S$ is a component of $T$  $A \subseteq S$ is a quasicomponent of $T$.
Irreducible Space is Locally Connected
Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Then $T$ is locally connected.
Quasicomponents and Path Components are Equal in Locally Path-Connected Space
Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected. Then $A \subseteq S$ is a path component of $T$  $A \subseteq S$ is a quasicomponent of $T$.