hcju/proofwikips huggingface dataset:
Definition:Semilattice Homomorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices. Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in S$: :$\map \phi {a \circ b} = \map \phi a * \map \phi b$ Then $\phi: \struct {S, \circ} \to \struct {T, *}$ is a semilattice homomorphism.
Definition:Subsemilattice
Let $\struct {R, \circ}$ be an algebraic structure with a binary operation. A '''subsemilattice of $\struct {R, \circ}$''' is a subset $S$ of $R$ such that $\struct {S, \circ_S}$ is a semilattice.
Definition:Atom (Physics)/Diameter
The diameter of an '''atom''' (considered approximately spherical) is between about $1$ and $5$ angstroms.
Definition:Atom (Physics)/Classical
The classical model of the '''atom''' is as a hard spherical body which is electrically neutral. In this model, '''atoms''' exert no forces on one another until they are brought into contact, at which point they cannot be brought any closer together, no matter how hard they are pushed to do so.
Definition:Gram-Equivalent
The '''gram-equivalent''' of a chemical element $E$ in a particular compound is the ratio of the mass of $E$ to another, with the mass of hydrogen taken as $1$.
Definition:Ion
An '''ion''' is an atom or a fragment of a molecule which has an electric charge on it through having either more or fewer electrons than protons.
Definition:Faraday (Unit of Charge)
The '''Faraday''' is defined as the amount of electric charge needed to release one gram-equivalent of a chemical element. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''Faraday''' }} {{eqn | r = 96 \, 485       | c = coulombs per mole: $\mathrm C \, \mathrm {mol}^{-1}$ }} {{end-eqn}}  === Symbol === {{:Definition:Faraday (Unit of Charge)/Symbol}} {{NamedforDef|Michael Faraday|cat = Faraday}}
Definition:Water
'''Water''' is the common term for the chemical substance whose molecules are composed of $2$ atoms of hydrogen and $1$ atom of oxygen. It can be expressed using the chemical formula $\mathrm H_2 \mathrm O$.
Definition:Homogeneous (Physics)/Warning
Just to specify that a body ''is made of the same substance throughout'' is '''not''' an adequate definition of '''homogeneous'''. For example, a column of air in the atmosphere is denser at the bottom than at the top. The fact that it is "all made of air" is one thing, but the air at the bottom can be distinguished from that higher up because the densities are different. Thus a column of air is '''not''' '''homogeneous'''.
Definition:Molecular Weight
Let $S$ be a substance. Let $S$ be homogeneous. The '''molecular weight''' of $S$ is defined as: :$W_S = m_S \times \dfrac {12} {m_C}$ where: :$m_S$ is the (arithmetic) mean mass of one molecule of $S$ :$m_C$ is the mass of one molecule of carbon-12. It is understood that the atoms that compose the molecules of $S$ may be of different isotopes according to their relative natural concentrations at the source of wherever $S$ came from. Hence the use of the (arithmetic) mean mass in the definition of $m_S$.
Definition:Avogadro's Number
'''Avogadro's number''' is defined as the number of particles in a mole. Its value is defined as: :$6 \cdotp 02214 076 \times 10^{23}$ and is usually denoted either $N_A$ or $L$. Thus it is the constant of proportion for converting the mass of one mole of a substance to the (arithmetic) mean mass of one molecule of that substance. {{NamedforDef|Lorenzo Romano Amedeo Carlo Avogadro|cat = Avogadro}}
Definition:Fiber of Truth/Solution
Let $P: X \to \set {\T, \F}$ be a propositional function defined on a domain $X$. Let $S = \set {x \in X: \map P x = \T}$ be the '''fiber of truth (under $P$)'''. Then an element of $S$ is known as a '''solution''' of $P$.
Definition:Partial Derivative/Value at Point
Let $\map f {x_1, x_2, \ldots, x_n}$ be a real function of $n$ variables Let $f_i = \dfrac {\partial f} {\partial x_i}$ be the partial derivative of $f$ {{WRT|Differentiation}} $x_i$. Then the value of $f_i$ at $x = \tuple {a_1, a_2, \ldots, a_n}$ can be denoted: :$\valueat {\dfrac {\partial f} {\partial x_i} } {x_1 \mathop = a_1, x_2 \mathop = a_2, \mathop \ldots, x_n \mathop = a_n}$ or: :$\valueat {\dfrac {\partial f} {\partial x_i} } {a_1, a_2, \mathop \ldots, a_n}$ or: :$\map {f_i} {a_1, a_2, \mathop \ldots, a_n}$ and so on. Hence we can express: :$\map {f_i} {a_1, a_2, \mathop \ldots, a_n} = \valueat {\dfrac \partial {\partial x_i} \map f {a_1, a_2, \mathop \ldots, a_{i - 1}, x_i, a_{i + i}, \mathop \ldots, a_n} } {x_i \mathop = a_i}$ according to what may be needed as appropriate.
Definition:Closure (Topology)/Definition 6
The '''closure of $H$ (in $T$)''', denoted $H^-$, is the set of all adherent points of $H$.
Definition:Partial Derivative/Higher Derivative/Third Derivative
Let $u = \map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$. The following is an example of one of the $3$rd derivatives of $f$: :$\dfrac {\partial^3 u} {\partial z^2 \partial y} := \map {\dfrac \partial {\partial z} } {\dfrac {\partial^2 u} {\partial z \partial y} } =: \map {f_{2 3 3} } {x, y, z}$
Definition:Partial Derivative/Higher Derivative/Fourth Derivative
Let $u = \map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$. The following is an example of one of the $4$th derivatives of $f$: :$\dfrac {\partial^4 u} { \partial x \partial y \partial z^2} := \map {\dfrac \partial {\partial x} } {\dfrac {\partial^3 u} {\partial y \partial z^2} } =: \map {f_{3 3 2 1} } {x, y, z}$
Definition:Grandi's Series
'''Grandi's series''' is the divergent series: :$\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n$ {{NamedforDef|Luigi Guido Grandi|cat = Grandi}}
Definition:Partial Derivative/Order
$u = \map f {x_1, x_2, \ldots, x_n}$ be a function of the $n$ independent variables $x_1, x_2, \ldots, x_n$. The '''order''' of a partial derivative of $u$ is the '''number of times it has been (partially) differentiated''' by at least one of $x_1, x_2, \ldots, x_n$. For example: :a second partial derivative of $u$ is of '''second order''', or '''order $2$''' :a third partial derivative of $u$ is of '''third order''', or '''order $3$''' and so on.
Definition:Event/Occurrence/Impossibility
Let $A \in \Sigma$ be an event of $\EE$ whose probability of occurring is equal to $0$. Then $A$ is described as '''impossible'''. That is, it is an '''impossibility''' for $A$ to occur.
Definition:Event/Occurrence/Certainty
Let $A \in \Sigma$ be an event of $\EE$ whose probability of occurring is equal to $1$. Then $A$ is described as '''certain'''. That is, it is a '''certainty''' that $A$ occurs.
Definition:Event/Occurrence/Equality
Let $A, B \in \Sigma$ be events of $\EE$ such that $A = B$. Then: :the occurrence of $A$ inevitably brings about the occurrence of $B$ and: :the occurrence of $B$ inevitably brings about the occurrence of $A$.
Definition:Event/Occurrence/Intersection
Let $\omega \in A \cap B$, where $A \cap B$ denotes the intersection of $A$ and $B$. Then '''both $A$ and $B$ occur'''.
Definition:Event/Occurrence/Union
Let $\omega \in A \cup B$, where $A \cup B$ denotes the union of $A$ and $B$. Then '''either $A$ or $B$ occur'''.
Definition:Event/Occurrence/Difference
Let $\omega \in A \setminus B$, where $A \setminus B$ denotes the difference of $A$ and $B$. Then '''$A$ occurs but $B$ does not occur'''.
Definition:Event/Occurrence/Symmetric Difference
Let $\omega \in A * B$, where $A * B$ denotes the symmetric difference of $A$ and $B$. Then '''either $A$ occurs or $B$ occurs, but not both'''.
Definition:Uniform Cauchy Criterion
Let $S \subseteq \mathbb R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.  We say that $\sequence {f_n}$ satisfies the '''uniform Cauchy criterion''' or is '''uniformly Cauchy''' on $S$ if for all $\varepsilon \in \R_{> 0}$, there exists $N \in \N$ such that:  :$\size {\map {f_n} x - \map {f_m} x} < \varepsilon$ for all $x \in S$ and $n, m > N$. By Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent, this criterion gives a necessary and sufficient condition for a sequence of real functions to be uniformly convergent.
Definition:Complete Set of Events
Let $I$ be an indexing set. Let $\family {A_i}_{i \mathop \in I}$ be a family of events in a probability space indexed by $I$. $\family {A_i}_{i \mathop \in I}$ is a '''complete set of events''' : :$\displaystyle \map \Pr {\bigcup_{i \mathop \in I} A_i} = 1$
Definition:Adherent Point/Definition 3
A point $x \in S$ is an '''adherent point of $A$'''  every neighborhood $N$ of $x$ satisfies: :$A \cap N \ne \O$
Definition:Boundary (Topology)/Definition 1
The '''boundary of $H$''' consists of all the points in the closure of $H$ which are not in the interior of $H$. Thus, the '''boundary of $H$''' is defined as: :$\partial H := H^- \setminus H^\circ$ where $H^-$ denotes the closure and $H^\circ$ the interior of $H$.
Definition:Boundary (Topology)/Definition 2
$x \in S$ is a '''boundary point''' of $H$ if every neighborhood $N$ of $x$ satisfies: :$H \cap N \ne \O$ and :$\overline H \cap N \ne \O$ where $\overline H$ is the complement of $H$ in $S$. The '''boundary of $H$''' consists of all the '''boundary points''' of $H$.
Definition:Set Meeting Set
Let $\family {S_i}_{i \mathop \in I}$ be an family of sets indexed by some  indexing set $I$. The sets in $\family {S_i}$ are said to '''meet'''  their intersection is not empty. That is, : :$\displaystyle \bigcap_{i \mathop \in I} \family {S_i} \ne \O$ That is,  $\family {S_i}_{i \mathop \in I}$ is not disjoint.
Definition:Disjoint Sets/Family
Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Then $\family {S_i}_{i \mathop \in I}$ is '''disjoint'''  their intersection is empty: :$\displaystyle \bigcap_{i \mathop \in I} S_i = \O$
Definition:Negative Vector
The inverse of a vector $\mathbf x \in G$ is usually denoted $-\mathbf x$, and called the '''negative of $\mathbf x$'''.
Definition:Vector Subtraction
Let $\struct {F, +_F, \times_F}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $V := \struct {G, +_G, \circ}_R$ be the corresponding '''vector space over $F$'''. Let $\mathbf x$ and $\mathbf y$ be vectors of $V$. Then the operation of '''(vector) subtraction''' on $\mathbf x$ and $\mathbf y$ is defined as: :$\mathbf x - \mathbf y := \mathbf x + \paren {-\mathbf y}$ where $-\mathbf y$ is the negative of $\mathbf y$. The $+$ on the  is vector addition. === Arrow Representation === {{:Definition:Vector Subtraction/Arrow Representation}}
Definition:Bounded Variation
Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a real function.  For each finite subdivision $P$ of $\closedint a b$, write:  :$P = \set {x_0, x_1, \ldots, x_n}$ with: :$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$ Also write: :$\displaystyle \map {V_f} P = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$ We say $f$ is of '''bounded variation''' if there exists a $M \in \R$ such that: :$\map {V_f} P \le M$ for all finite subdivisions $P$.
Definition:Total Variation
Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a function of bounded variation. Let $X$ be the set of finite subdivisions of $\closedint a b$. For each $P \in X$, write:  :$P = \set {x_0, x_1, \ldots, x_{\size P - 1} }$ with: :$a = x_0 < x_1 < x_2 < \cdots < x_{\size P - 2} < x_{\size P - 1} = b$ Also write: :$\displaystyle \map {V_f} P = \sum_{i \mathop = 1}^{\size P - 1} \size {\map f {x_i} - \map f {x_{i - 1} } }$ We define the '''total variation''' $V_f$ of $f$ on $\closedint a b$ by:  :$\displaystyle V_f = \sup_{P \in X} \paren {\map {V_f} P}$ This supremum is finite as, since $f$ is of bounded variation, there exists $M \in \R$ with: :$\map {V_f} P \le M$ for all $P \in X$.
Definition:Truncated Real-Valued Function
Let $f: \R \to \R$ be a real function. Let $m \in \Z_{>0}$ be a (positive) integer constant. Let $f_m: \R \to \R$ be the real function defined as: :$\forall x \in \Dom f: \map {f_m} x = \begin {cases} \map f x & : \map f x \le m \\ m & : \map f x > m \end {cases}$ Then $\map {f_m} x$ is called '''(the function) $f$ truncated by $m$'''.
Definition:Integrable Function/Unbounded Above
Let $f: \R \to \R$ be a positive real function. Let $f$ be unbounded above on the open interval $\openint a b$. Let $f_m$ denote the function $f$ truncated by $m$ for $m \in \Z_{>0}$. Suppose that $f_m$ is Darboux integrable on $\openint a b$ for all $m \in \Z_{>0}$. Suppose also that the following limit exists: :$\displaystyle \lim_{m \mathop \to \infty} \int_a^b \map {f_m} x \rd x$ Then $f$ is '''integrable on $\openint a b$''' and can be expressed in the conventional notation of the Darboux integral: :$\displaystyle \int_a^b \map f x \rd x$
Definition:Integrable Function/Unbounded
Let $f: \R \to \R$ be a real function. Let $f$ be unbounded on the open interval $\openint a b$. Let: :$f^+$ denote the positive part of $f$ :$f^-$ denote the negative part of $f$. Let $f^+$ and $-f^-$ both be integrable on $\openint a b$. Then $f$ is '''integrable on $\openint a b$ and its (definite) integral is understood to be: :$\displaystyle \int_a^b \map f x \rd x := \int_a^b \map {f^+} x \rd x - \int_a^b \paren {-\map {f^-} x} \rd x$
Definition:Integrable Function/Measure Space
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function. Then $f$ is said to be '''$\mu$-integrable''' : :$\displaystyle \int f^+ \rd \mu < +\infty$ and :$\displaystyle \int f^- \rd \mu < +\infty$ where $f^+$, $f^-$ are the positive and negative parts of $f$, respectively. The integral signs denote $\mu$-integration of positive measurable functions.
Definition:Baire Category
The '''Baire category''' of a topological space $T$ is a way of specifying how dense $T$ is: === First Category (Meager) === {{:Definition:Meager Space}} === Second Category (Non-Meager) === {{:Definition:Non-Meager Space}} {{NamedforDef|René-Louis Baire|cat = Baire}}
Definition:Homeomorphism/Terminology
Let a '''homeomorphism''' exist between $T_\alpha$ and $T_\beta$. Then $T_\alpha$ and $T_\beta$ are said to be '''homeomorphic'''. The symbolism $T_\alpha \sim T_\beta$ is often seen to denote that $T_\alpha$ is '''homeomorphic''' to $T_\beta$.
Definition:Filter Basis/Definition 1
Let $\BB \subset \powerset S$ such that $\O \notin \BB$ and $\BB \ne \O$. Then $\FF := \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ : :$\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$ Such a $\BB$ is called a '''filter basis''' of $\FF$.
Definition:Filter Basis/Generated Filter
$\FF$ is said to be '''generated by $\BB$'''.
Definition:Filter Basis/Definition 2
Let $\BB$ be a subset of a filter $\FF$ on $S$ such that $\BB \ne \O$. Then $\BB$ is a '''filter basis''' of $\FF$ : :$\forall U \in \FF: \exists V \in \BB: V \subseteq U$
Definition:Fleenor-Heronian Triangle
A '''Fleenor-Heronian triangle''' is a '''Heronian triangle''' whose sides have lengths form a set of $3$ consecutive integers.
Definition:Fleenor-Heronian Triangle/Sequence
The sequence of Fleenor-Heronian triangles begins: :$\paren {1, 2, 3}$ :$\paren {3, 4, 5}$ :$\paren {13, 14, 15}$ :$\paren {51, 52, 53}$ :$\paren {193, 194, 195}$ :$\paren {723, 724, 725}$ :$\paren {2701, 2702, 2703}$ :$\paren {10 \, 083, 10 \, 084, 10 \, 085}$ :$\paren {37 \, 633, 37 \, 634, 37 \, 635}$
Definition:Countable Topological Space
A '''countable topological space''' is a '''topological space''' whose underlying set is '''countably infinite'''.
Definition:Uncountable Topological Space
An '''uncountable topological space''' is a '''topological space''' whose underlying set is '''uncountable'''.
Definition:Finite Topological Space
A '''finite topological space''' is a '''topological space''' whose underlying set is '''finite'''.
Definition:Uncountable Discrete Ordinal Space
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Let $S$ be the set of points of $\hointr 0 \Omega$ of the form $\alpha + 1$, where $\alpha$ is a limit ordinal. Let $\tau$ be the subspace topology induced by the order topology on $\hointr 0 \Omega$. $\struct {S, \tau}$ is known as the '''uncountable discrete ordinal space'''.
Definition:Long Line
Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Let $L$ be the set constructed as follows. Between each ordinal $\alpha \in \hointr 0 \Omega$ and its successor $\alpha + 1$, let a copy of the open (real) unit interval $\openint 0 1$ be inserted. Let a total ordering $\preccurlyeq$ be applied to $L$ according to the betweenness described above. Let the order topology $\tau$ be applied to the ordered structure $\struct {L, \preccurlyeq}$. The resulting topological space $\struct {L, \preccurlyeq, \tau}$ is called the '''long line'''. Informally it can be seen that $L$ is of the form: :$0, \openint 0 1, 1, \openint 0 1, 2, \openint 0 1, \ldots, \openint 0 1, \alpha, \openint 0 1, \alpha + 1, \openint 0 1, \ldots, \openint 0 1, \Omega - 1, \openint 0 1$
Definition:Extended Long Line
Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the open ordinal space on $\Omega$. Let $L^*$ be the set constructed as follows. Between each ordinal $\alpha \in \closedint 0 \Omega$ and its successor $\alpha + 1$, let a copy of the open (real) unit interval $\openint 0 1$ be inserted. Let a total ordering $\preccurlyeq$ be applied to $L^*$ according to the betweenness described above. Let the order topology $\tau$ be applied to the ordered structure $\struct {L^*, \preccurlyeq}$. The resulting topological space $\struct {L^*, \preccurlyeq, \tau}$ is called the '''extended long line'''. Informally it can be seen that $L$ is of the form: :$0, \openint 0 1, 1, \openint 0 1, 2, \openint 0 1, \ldots, \openint 0 1, \alpha, \openint 0 1, \alpha + 1, \openint 0 1, \ldots, \openint 0 1, \Omega - 1, \openint 0 1, \Omega$
Definition:Altered Long Line
Let $\struct {L, \preccurlyeq, \tau}$ denote the '''long line'''. Let $p$ be a point which is not an element of $L$. Consider the set $S := L \cup \set p$. Let $T = \struct {S, \tau_p}$ be the topological space whose topology $\tau_p$ is defined as follows: The open sets of $T$ are the open sets of $L$ together with those generated by the following neighborhoods of $P$: :$\map {U_s} p := \set p \cup \set {\displaystyle \bigcup_{\alpha \mathop = \beta}^\Omega \openint \alpha {\alpha + 1}: 1 \le \beta < \Omega}$ where $\Omega$ denotes the first uncountable ordinal. $\map {U_s} p$ is therefore a right-hand ray without the ordinals. The topological space $T = \struct {S, \tau_p}$ is known as the '''altered long line'''.
Definition:Altered Long Line/Greatest Element
$p$ is the greatest element of $S = L \cup \set p$.
Definition:Unit Square under Lexicographic Ordering
Let $S$ be the unit square in the (real) Cartesian plane: {{begin-eqn}} {{eqn | l = S       | r = \set {\tuple {x, y}: x, y \in \R, 0 \le x \le 1, 0 \le y \le 1}       | c =  }} {{eqn | r = \closedint 0 1 \times \closedint 0 1       | c = where $\closedint 0 1$ is the closed unit interval }} {{end-eqn}} Let $\preccurlyeq$ be the lexicographical ordering applied to $S$: :$\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S: \tuple {x_1, y_1} \preccurlyeq \tuple {x_2, y_2} \iff \begin {cases} x_1 < x_2 \\ x_1 = x_2 \land y_1 \le y_2 \end {cases}$ Let $\tau$ be the order topology be applied to the ordered structure $\struct {S, \preccurlyeq}$. Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the '''unit square under lexicographical ordering'''.
Definition:Right Order Topology
Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $\tau$ be the topology on $S$ generated by the basis sets of the form: :$S_a = \set {x: a \prec x}$ for $a \in S$. Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the '''right order topology on $S$'''.
Definition:Left Order Topology
Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $\tau$ be the topology on $S$ generated by the basis sets of the form: :$S_a = \set {x: x \prec a}$ for $a \in S$. Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the '''left order topology on $S$'''.
Definition:Nested Interval Topology
Let $S = \openint 0 1$ denote the open real interval: :$\openint 0 1 = \set {x \in \R: 0 < x < 1}$ Let $\tau$ be the topology defined on $\openint 0 1$ by defining the open sets $U_n$ as: :$\forall n \in \N_{>0}: U_n := \openint 0 {1 - \dfrac 1 n}$ together with $\O$ and $S$ itself. Then $\tau$ is referred to as the '''nested interval topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''nested interval space'''.
Definition:Overlapping Interval Topology
Let $S = \closedint {-1} 1$ denote the open real interval: :$\closedint {-1} 1 = \set {x \in \R: -1 \le x \le 1}$ Let $\BB$ be the set: :$\BB = \set {\hointl a 1: -1 < a < 0} \cup \set {\hointr {-1} b: 0 < b < 1}$ where: :$\hointl a 1$ is the half-open interval $\set {x \in \R: a < x \le 1}$. :$\hointr {-1} b$ is the half-open interval $\set {x \in \R: -1 \le x < b}$. Then $\BB$ is the basis for a topology $\tau$ on $\R$. Thus the sets of the form $\openint a b$ such that $a < 0 < b$ are open sets in $S$. $\tau$ is referred to as the '''overlapping interval topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''overlapping interval space'''.
Definition:Hjalmar Ekdal Topology
Let $\tau$ be the topology defined on the (strictly) positive integers $\Z_{>0}$ as follows: A subset $H$ of $\Z_{>0}$ is in $\tau$  $H$ contains the (immediate) successor of every odd integer that is also in $H$: :$\tau := \set {H \subseteq \Z_{>0}: 2 n - 1 \in H \implies 2 n \in H}$ Then $\tau$ is referred to as the '''Hjalmar Ekdal topology'''. The topological space $T = \struct {\Z_{>0}, \tau}$ is referred to as the '''Hjalmar Ekdal space'''.
Definition:Prime Ideal Topology
Let $S$ be the set of all prime ideals $P$ of the integers $\Z$. Let $\BB$ be the set of all sets $V_x$ defined as: :$V_x = \set {P \in S: x \notin P}$ for all $x \in \Z_{>0}$. Then $\BB$ is the basis for a topology $\tau$ on $S$. $\tau$ is referred to as the '''prime ideal topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''prime ideal space'''.
Definition:Divisor Topology
Let $S = \set {x \in \Z: x \ge 2}$ denote the set of integers greater than $1$. Let $\BB$ be the set of all sets $U_n$ defined for all $n \ge 2$ as: :$U_n = \set {x \in \Z_{>0}: x \divides n}$ where $\divides$ denotes the divisor relation. Then $\BB$ is the basis for a topology $\tau$ on $S$. Then $\tau$ is referred to as the '''divisor topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''divisor space'''.
Definition:Evenly Spaced Integer Topology
Let $\Z$ denote the set of integers. Let $\BB$ be the set of sets defined as: :$\BB = \set {a + k \Z: a, k \in \Z, k \ne 0}$ where $a + k \Z := \set {a + k \lambda: \lambda \in \Z}$. Then $\BB$ is the basis for a topology $\tau$ on $S$. Then $\tau$ is referred to as the '''evenly spaced integer topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''evenly spaced integer space'''.
Definition:P-adic Topology
Let $\Z$ denote the set of integers. Let $p \in \Z$ be a fixed prime number. Let $\BB$ be the set of all sets $\map {U_\alpha} n$ defined as: :$\map {U_\alpha} n = \set {n + \lambda p^\alpha: \lambda \in \Z}$ Then $\BB$ is the basis for a topology $\tau$ on $S$. $\tau$ is referred to as the '''$p$-adic topology (on $\Z$)'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''$p$-adic (topological) space'''.