nyuuzyou/wikis huggingface dataset:
Characterization of Paracompactness in T3 Space/Lemma 9

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological space.
Let every open cover of $T$ be even.
Let $\BB$ be a discrete set of subsets of $X$.
Then there exists an open neighborhood $W$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:
:$\forall x \in X : \card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$
== Proof ==
Let:
:$\UU = \set{ U \in \tau : \card {\set{B \in \BB : U \cap B} } \le 1}$
==== Lemma 19 ====
We have :
:$\UU$ is an even cover.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
By definition of even cover, there exists a neighborhood $V$ of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$:
:$\set{\map V x : x \in X}$ is a refinement of $\UU$
==== Lemma 20 ====
From Lemma 20, there exists an open neighborhood $W$ of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:
:$W = W^{-1}$
:$W \circ W \subseteq V$
==== Lemma 21 ====
Let $x \in X$.
By definition of refinement:
:$\exists U \in \UU : \map V x \subseteq U$
From Corollary to Image under Subset of Relation is Subset of Image under Relation:
:$\map {W \circ W} x \subseteq \map V x$
From Subset Relation is Transitive:
:$\map {W \circ W} x \subseteq U$
We have:
By definition of $\UU$:
:$\card {\set{B \in \BB : U \cap B \ne \O}} \le 1$
Hence:
:$\card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$
Since $x$ was arbitrary, it follows that:
:$\forall x \in X : \card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological space.
If $T$ is paracompact then:
:every open cover of $T$ has a locally finite refinement
== Proof ==
Let $T$ be paracompact.
By definition of paracompact:
:every open cover of $S$ has an open refinement which is locally finite.
By definition of open refinement:
:every open refinement of a cover is a refinement of the cover.
It follows that:
:every open cover of $T$ has a locally finite refinement.
== Sources ==
* : Chapter $5$: Compact Spaces: $\S$ Paracompactness: Theorem $28$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 6

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological space.
If $T$ is paracompact then:
:every open cover of $T$ has an open $\sigma$-locally finite refinement
== Proof ==
Let $T$ be paracompact.
By definition of paracompact:
:every open cover of $T$ has an open locally finite refinement
From Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets
:every open cover of $T$ has an open $\sigma$-locally finite refinement
== Sources ==
* : Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 2 implies Statement 3

== Theorem ==
Let $T = \struct{X, \tau}$ be a $T_3$ space.
If every open cover of $T$ has a locally finite refinement then:
:every open cover of $T$ has a closed locally finite refinement
== Proof ==
Let every open cover of $T$ have a locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$ where $V^-$ denotes the closure of $V$ in $T$.
==== Lemma 1 ====
By assumption:
:there exists a locally finite refinement $\AA$ of $\VV$.
Let:
:$\BB = \set{A^- : A \in \AA}$
From Closures of Elements of Locally Finite Set is Locally Finite:
:$\BB$ is locally finite
==== Lemma 2 ====
==== Lemma 3 ====
Since $\UU$ was an arbitrary open cover of $T$ it follows that:
:every open cover of $T$ has a closed locally finite refinement.
== Sources ==
* : Chapter $5$: Compact Spaces: $\S$Paracompactness: Lemma $29$
* : Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 1

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ have a closed locally finite refinement then:
:$T$ is paracompact.
== Proof ==
Let every open cover of $T$ have a closed locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV$ be a closed locally finite refinement of $\UU$, which exists by assumption.
Let $\WW = \set{W \in \tau : \set{V \in \VV : V \cap W \ne \O} \text{ is finite}}$.
By definition of locally finite:
:$\forall x \in X: \exists W \in \tau: x \in W$ and $\set{V \in \VV : V \cap W \ne \O}$ is finite. 
Hence $\WW$ is an open cover of $T$, by definition.
Let $\AA$ be a closed locally finite refinement of $\WW$, which exists by assumption.
==== Lemma 4 ====
For each $V \in \VV$, let:
:$V^* = X \setminus \ds \bigcup \set{A \in \AA : A \cap V = \O}$
==== Lemma 5 ====
Let $\VV^* = \set{V^* : V \in \VV}$.
==== Lemma 6 ====
From Lemma 5 and Lemma 6 it follows that $\VV$ is a refinement of $\VV^*$ by definition.
By definition of refinement:
:$\forall V \in \VV : \exists U \in \UU : V \subseteq U$
For each $V \in \VV$, let:
:$U_V \in \UU : V \subseteq U_V$
Let:
:$\UU^* = \set{V^* \cap U_V : V \in \VV}$
==== Lemma 7 ====
Since $\UU$ was arbitrary, it follows that $T$ is paracompact by definition.
== Sources ==
* : Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 4

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has a closed locally finite refinement then:
:every open cover of $T$ is even
== Proof ==
This follows immediately from Open Cover with Closed Locally Finite Refinement is Even Cover.
== Sources ==
* : Chapter $5$: Compact Spaces: $\S$Paracompactness: Theorem $28$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 4 implies Statement 5

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological Space.
If every open cover of $T$ is even then:
:every open cover of $T$ has an open $\sigma$-discrete refinement
== Proof ==
Let every open cover of $T$ be even.
Let $\UU$ be an open cover of $T$.
==== Lemma 8 ====
By definition of $\sigma$-discrete set of subsets:
:$\AA = \ds \bigcup_{n \in \N} \AA_n$ where $\AA_n$ is a discrete set of subsets for each $n \in \N$.
Let $X \times X$ denote the cartesian product of $X$ with itself.
Let $\tau_{X \times X}$ denote the product topology on $X \times X$.
Let $T \times T$ denote the product space $\struct {X \times X, \tau_{X \times X} }$. 
==== Lemma 9 ====
From Lemma 9:
:$\forall n \in \N$ there exists a neighborhood $V_n$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:
::$\forall x \in X : \card {\set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \ne \O}} \le 1$
By definition of refinement:
:$\forall A \in \AA : \exists U_A \in U : A \subseteq U_A$
For each $n \in \N$, let:
:$\WW_n = \set{U_A \cap V_n \sqbrk A : A \in \AA_n}$
Let:
:$\WW = \ds \bigcup_{n \in \N} \WW_n$
==== Lemma 10 ====
Since $\UU$ was arbitrary, then every open cover of $T$ has an open $\sigma$-discrete refinement
== Sources ==
* : Chapter $5$: Compact Spaces: $\S$Paracompactness: Lemma $33$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 5 implies Statement 6

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has an open $\sigma$-discrete refinement then:
:every open cover of $T$ has an open $\sigma$-locally finite refinement
== Proof ==
Follows immediately from Sigma-Discrete Set of Subsets is Sigma-Locally Finite.
== Sources ==
* : Chapter $5$: Compact Spaces: $\S$Paracompactness: Theorem $28$
Category:Characterization of Paracompactness in T3 Space
Characterization of Paracompactness in T3 Space/Statement 6 implies Statement 2

== Theorem ==
Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has an open $\sigma$-locally finite refinement then:
:every open cover of $T$ has a locally finite refinement
== Proof ==
Let every open cover of $T$ have an open $\sigma$-locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV$ be an open $\sigma$-locally finite refinement of $\UU$ .
From Sigma-Locally Finite Cover has Locally Finite Refinement:
:there exists a locally finite refinement $\AA$ of $\VV$
From Refinement of a Refinement is Refinement of Cover:
:$\AA$ is a locally finite refinement of $\UU$
Since $\UU$ was arbitrary, it follows that:
:every open cover of $T$ has a locally finite refinement
== Sources ==
* : Chapter $5$: Compact Spaces: $\S$Paracompactness: Lemma $34$
* : Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$
Category:Characterization of Paracompactness in T3 Space
Characterization of Pointwise Maximum of Real-Valued Functions

== Theorem ==
Let $S$ be a set.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be real-valued functions.
Let $f \vee g$ denote the pointwise maximum of $f$ and $g$, that is, $f \vee g$ is the mapping defined by:
:$\forall s \in S : \map {\paren{f \vee g} } s = \max \set{\map f s, \map g s}$
Then:
:$f \vee g = \dfrac 1 2 \paren{f + g + \size{f - g}}$
where:
:$f + g$ denotes the pointwise addition of $f$ and $g$
:$f - g$ denotes the pointwise difference of $f$ and $g$
:$\size{f - g}$ denotes the absolute value of $f - g$
:$\dfrac 1 2 \paren{f + g + \size{f - g}}$ denotes the pointwise scalar multiplication of $f + g + \size{f - g}$ by $\dfrac 1 2$
== Proof ==
We have:
By definition of equality of mappings:
:$f \vee g = \dfrac 1 2 \paren{f + g + \size{f - g}}$
Category:Real-Valued Functions
Characterization of Pointwise Minimum of Real-Valued Functions

== Theorem ==
Let $S$ be a set.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be real-valued functions.
Let $f \wedge g$ denote the pointwise maximum of $f$ and $g$, that is, $f \wedge g$ is the mapping defined by:
:$\forall s \in S : \map {\paren{f \wedge g} } s = \min \set{\map f s, \map g s}$
Then:
:$f \wedge g = \dfrac 1 2 \paren{f + g - \size{f - g}}$
where:
:$f + g$ denotes the pointwise addition of $f$ and $g$
:$f - g$ denotes the pointwise difference of $f$ and $g$
:$\size{f - g}$ denotes the absolute value of $f - g$
:$\dfrac 1 2 \paren{f + g - \size{f - g}}$ denotes the pointwise scalar multiplication of $f + g - \size{f - g}$ by $\dfrac 1 2$
== Proof ==
We have:
By definition of equality of mappings:
:$f \wedge g = \dfrac 1 2 \paren{f + g - \size{f - g}}$
Category:Real-Valued Functions
Characterization of Polynomial has Root in P-adic Integers

== Theorem ==
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a polynomial over $\Z_p$.
Let $a \in \Z_p$.
Then:
:$\map F a = 0$
:
:there exists a sequence $\sequence {a_n}$ of integers: 
::$(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
::$(2): \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$
where $\map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$ denotes congruence modulo the ideal $p^{n + 1} \Z_p$
== Proof ==
=== Necessary Condition ===
=== Sufficient Condition ===
Category:P-adic Integers
Category:Characterization of Polynomial has Root in P-adic Integers
Characterization of Polynomial has Root in P-adic Integers/Necessary Condition

== Theorem ==
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a polynomial over $\Z_p$.
Let $a \in \Z_p$.
Let $\map F a = 0$.
Then:
:there exists a sequence $\sequence {a_n}$ of integers: 
::$(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
::$(2): \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$
where $\map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$ denotes congruence modulo the ideal $p^{n + 1} \Z_p$
== Proof ==
Let $\map F a = 0$.
Let $a = \ds \sum_{j \mathop = 0}^\infty d_j p^j$ be the $p$-adic expansion of $a$.
For all $n \in \N_{>0}$, let:
:$a_n = \ds \sum_{j \mathop = 0}^{n - 1} d_j p^j$
By definition of $p$-adic expansion:
:$\ds \lim_{n \mathop \to \infty} {a_n} = a$
By definition of $p$-adic expansion of a $p$-adic integer:
:$\forall n \in \N_{>0} : a_n \in \Z$
We have:
Category:Characterization of Polynomial has Root in P-adic Integers
Characterization of Polynomial has Root in P-adic Integers/Sufficient Condition

== Theorem ==
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a polynomial over $\Z_p$.
Let $a \in \Z_p$.
Let there exist a sequence $\sequence{a_n}$ of integers: 
:$(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
:$(2): \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$
where $\map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$ denotes congruence modulo the ideal $p^{n + 1} \Z_p$.
Then:
:$\map F a = 0$
== Proof ==
Let there exist a sequence $\sequence{a_n}$ of integers: 
:$(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
:$(2): \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$
We have:
Category:Characterization of Polynomial has Root in P-adic Integers
Characterization of Positive Bounded Linear Operator on Hilbert Space

== Theorem ==
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B \HH$ be the space of bounded linear operators on $\HH$ understood as a $\text C^\ast$-algebra.
Let $T : \HH \to \HH$ be a bounded linear operator.
Then $T$ is positive in $\map B \HH$ :
:for each $x \in \HH$, $\innerprod {T x} x \in \R_{\ge 0}$.
== Proof ==
=== Necessary Condition ===
Suppose that $T$ is positive in $\map B \HH$.
From Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra, there exists a Hermitian operator $R$ such that:
:$T = R^2$
Since $R$ is Hermitian we have:
:$R^2 = R^\ast R$ 
Then for each $x \in \HH$ we have:
=== Sufficient Condition ===
Suppose:
:for each $x \in \HH$, $\innerprod {T x} x \in \R_{\ge 0}$.
From Operator is Hermitian iff Numerical Range is Real:
:$T$ is Hermitian.
We show that $\map \sigma T \subseteq \hointr 0 \infty$.
Let $\lambda \in \map \sigma T$.
From Characterization of Spectrum of Bounded Linear Operator in Hilbert Space in terms of Approximate Eigenvalues: Corollary:
:$\lambda$ is an approximate eigenvalue of $T$.
Then there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ with $\norm {x_n} = 1$ for each $n \in \N$ such that:
:$\norm {T x_n - \lambda x_n} \to 0$
From Cauchy-Bunyakovsky-Schwarz Inequality, we have:
:$\cmod {\innerprod {T x_n - \lambda x_n} {x_n} } \le \norm {T x_n - \lambda x_n}$
So:
:$\innerprod {T x_n - \lambda x_n} {x_n} \to 0$ as $n \to \infty$
We have:
:$\innerprod {T x_n - \lambda x_n} {x_n} = \innerprod {T x_n} {x_n} - \lambda \norm {x_n}^2 = \innerprod {T x_n} {x_n} - \lambda$
from the definition of the inner product norm. 
So we have:
:$\innerprod {T x_n} {x_n} \to \lambda$ as $n \to \infty$
Since $\innerprod {T x_n} {x_n} \in \R_{\ge 0}$ for each $n \in \N$ and $\R_{\ge 0}$ is closed in $\C$, we have $\lambda \in \R_{\ge 0}$. 
That is, $\map \sigma T \subseteq \hointr 0 \infty$.
Category:Positive Elements of C*-Algebras
Characterization of Positive Element of Unital C*-Algebra

== Theorem ==
Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $a \in A$ be Hermitian. 
Let $t \ge 0$. 
Then if $\norm {a - t {\mathbf 1}_A} \le t$, $a$ is positive.
Conversely if $\norm a \le t$ and $a$ is positive, then $\norm {a - t {\mathbf 1}_A} \le t$. 
== Proof ==
Let $B$ to be $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x}$. 
By C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative, $B$ is commutative.
Further from Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra we have $\map {\sigma_B} x = \map {\sigma_A} x$.
Let $\Phi_B$ be the spectrum of $B$.
From Gelfand-Naimark Theorem: Commutative Case: Unital, there exists an isometric unital $\ast$-algebra isomorphism $G : B \to \map \CC {\Phi_B}$.
Suppose that $\norm {a - t {\mathbf 1}_A} \le t$.
Since $G$ is isometric and unital, we have:
:$\norm {a - t {\mathbf 1}_A} = \norm {\map G a - t}_\infty \le t$
where $\norm {\, \cdot \,}_\infty$ is the supremum norm on $\map \CC {\Phi_B}$.
Write $f = \map G a$ for brevity.
Since $G$ is a $\ast$-algebra isomorphism, $f$ is Hermitian.
Hence $\overline {\map f \phi} = \map f \phi$ for each $\phi \in \Phi_B$.
From Complex Number equals Conjugate iff Wholly Real, we then have $\map f \phi \in \R$ for each $\phi \in \Phi_B$.
Hence $\norm {f - t}_\infty \le t$ implies that $\map f \phi \in \closedint 0 {2 t}$ for each $\phi \in \Phi_B$.
From Spectrum of Element of Space of Continuous Functions on Compact Hausdorff Space, we therefore have $\map {\sigma_{\map \CC {\Phi_B} } } f \subseteq \closedint 0 {2 t}$.
Then, from Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism, we have:
:$\map {\sigma_A} a = \map {\sigma_B} a = \map {\sigma_{\map \CC {\Phi_B} } } f \subseteq \closedint 0 {2 t} \subseteq \hointr 0 \infty$
Hence $a$ is positive.
Now suppose that $\norm a \le t$ and $a$ is positive.
Again write $f = \map G a$. 
From Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism we have:
:$\map {\sigma_{\map \CC {\Phi_B} } } f = \map {\sigma_B} a = \map {\sigma_A} a$ 
So $\map {\sigma_{\map \CC {\Phi_B} } } f \subseteq \hointr 0 \infty$.
From Spectrum of Element of Space of Continuous Functions on Compact Hausdorff Space we therefore have:
:$\set {\map f \phi : \phi \in \Phi_B} \subseteq \hointr 0 \infty$
Since $G$ is isometric, we have $\norm f_\infty \le t$. 
So $\map f \phi \in \closedint 0 t$ for each $\phi \in \Phi_B$. 
Hence $t - \map f \phi \in \closedint 0 t$ for each $\phi \in \Phi_B$.
We therefore have $\norm {t - \map G a}_\infty \le t$. 
Again since $G$ is an isometric unital $\ast$-algebra isomorphism, we have $\norm {a - t {\mathbf 1}_A} \le t$. 
== Sources ==
* : $2.2$: Positive Elements of $C^\ast$-Algebras
Category:Positive Elements of C*-Algebras
Characterization of Pre-Measures

== Theorem ==
Let $X$ be a set, and let $\SS \subseteq \powerset X$ be a collection of subsets of $X$.
Let $\O \in \SS$.
Denote $\overline \R_{\ge 0}$ for the set of positive extended real numbers.
A mapping $\mu: \SS \to \overline \R_{\ge 0}$ is a pre-measure :
:$(1):\quad \map \mu \O = 0$
:$(2):\quad \mu$ is finitely additive
:$(3):\quad$ For every increasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\SS$, if $E_n \uparrow E$ for some $E \in \SS$, then:
::$\ds \map \mu E = \lim_{n \mathop \to \infty} \map \mu {E_n}$
where $E_n \uparrow E$ denotes limit of increasing sequence of sets.
Alternatively, and equivalently, $(3)$ may be replaced by either of:
:$(3'):\quad$ For every decreasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\SS$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow E$ for some $E \in \SS$, then:
::$\ds \map \mu E = \lim_{n \mathop \to \infty} \map \mu {E_n}$
:$(3):\quad$ For every decreasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\SS$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow \O$, then:
::$\ds \lim_{n \mathop \to \infty} \map \mu {E_n} = 0$
where $E_n \downarrow E$ denotes limit of decreasing sequence of sets.
== Proof ==
Category:Pre-Measures
Characterization of Preordered Vector Spaces

== Theorem ==
Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $\succeq$ be a preordering on $X$.
Then $\struct {X, \succeq}$ is a preordered vector space there exists a convex cone $P \subseteq X$ such that $\succeq$ is the preordering on $X$ induced by $P$.
== Proof ==
=== Necessary Condition ===
Suppose that $\struct {X, \succeq}$ is a preordered vector space.
Let:
:$P = \set {x \in X : x \succeq 0}$
Let $\succeq^P$ be the preordering on $X$ induced by $P$.
We want to show that, for $x, y \in X$, we have $x \succeq y$ $x \succeq^P y$.
Let $x, y \in X$.
By the definition of the preordering on $X$ induced by $P$, we have:
:$x \succeq^P y$ $x - y \in P$
By the definition of $P$:
:$x - y \in P$ $x - y \succeq 0$
Since $\struct {X, \succeq}$ is a preordered vector space, we obtain:
:$x - y \succeq 0$ $x \succeq y$
Hence $\succeq$ is the preordering on $X$ induced by $P$.
=== Sufficient Condition ===
Let $P \subseteq X$ be a convex cone.
Let $\succeq^P$ be the preordering on $X$ induced by $P$.
We show that $\struct {X, \succeq^P}$ is a preordered vector space.
We need to establish:
:$(1) \quad$ for all $x, x', y, y' \in X$ such that $x \succeq x'$ and $y \succeq y'$, we have $x + y \succeq^P x' + y'$
:$(2) \quad$ for all $\alpha \in \R_{\ge 0}$ and $x, x' \in X$ with $x \succeq^P x'$, we have $\alpha x \succeq^P \alpha x'$.
Let $x, y, x', y' \in X$ be such that $x \succeq^P x'$ and $y \succeq^P y'$.
Then $x - x' \in P$ and $y - y' \in P$.
From the definition of a convex cone, we have $\paren {x - x'} + \paren {y - y'} \in P$.
That is, $\paren {x + y} - \paren {x' + y'} \in P$.
So we have $x + y \succeq^P x' + y'$, establishing $(1)$.
Now let $\alpha \in \R_{\ge 0}$ and $x, x' \in X$ be such that $x \succeq^P x'$.
Then we have $x - x' \in P$.
Since $P$ is a cone and $\alpha \in \R_{\ge 0}$, we have $\alpha \paren {x - x'} \in P$.
That is, $\alpha x - \alpha x' \in P$.
So we have $\alpha x \succeq^P \alpha x'$.
This establishes $(2)$ and hence shows that $\struct {X, \succeq^P}$ is a preordered vector space.
== Sources ==
* : $1.1$: Basic notions
Category:Preordered Vector Spaces
Characterization of Prime Element in Inclusion Ordered Set of Topology

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $L = \struct {\tau, \preceq}$ be an inclusion ordered set of $\tau$.
Let $Z \in \tau$.
Then $Z$ is prime element in $L$ :
:$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$
== Proof ==
=== Sufficient Condition ===
Assume that
:$Z$ is prime element in $L$.
Let $X, T \in \tau$ such that:
:$X \cap Y \subseteq Z$
By Join and Meet in Inclusion Ordered Set of Topology and definition of inclusion ordered set:
:$X \wedge Y \preceq Z$
By definition of prime element:
:$X \preceq Z$ or $Y \preceq Z$
Thus by definition of inclusion ordered set:
:$X \subseteq Z$ or $Y \subseteq Z$
=== Necessary Condition ===
Assume that:
L$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$
Let $X, Y \in \tau$ such that:
:$X \wedge Y \preceq Z$
By Join and Meet in Inclusion Ordered Set of Topology and definition of inclusion ordered set:
:$X \cap Y \subseteq Z$
By assumption:
:$X \subseteq Z$ or $Y \subseteq Z$
Thus by definition of inclusion ordered set:
:$X \preceq Z$ or $Y \preceq Z$
== Sources ==
* 
Category:Prime Elements
Category:Topology