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- 10:08, 25 September 2022 Countably Additive Function Dichotomy by Empty Set (hist | edit) [1,335 bytes] Usagiop (talk | contribs) (Created page with "== Theorem == Let $\AA$ be a $\sigma$-algebra. Let $f: \AA \to \overline \R$ be a function, where $\overline \R$ denotes the extended set of real numbers. Let $f$ be a countably additive function. Then the one of the following is true: :$\paren 1$: $\map f \O = 0$ :$\paren 2$: $\map f A = + \infty$ for all $A \in \AA$ :$\paren 3$:...")
- 08:44, 25 September 2022 Inner Product/Examples/Lebesgue 2-Space (hist | edit) [2,899 bytes] Lord Farin (talk | contribs) (Created page with "{{rename|Given the importance of this inner product, it might need more exposure than 'just' an example}} == Example of Inner Product == <onlyinclude> Let $\tuple{ X, \Sigma, \mu }$ be a measure space. Let $\map {L^2} \mu$ be the Lebesgue $2$-space of $\mu$. Let $\innerprod \cdot \cdot: \map {L^2} \mu \times \map {L^2} \mu \to \C$ be the mapping defined by:...")
- 08:27, 25 September 2022 Euclidean Algorithm/Examples/341 and 527/Proof (hist | edit) [698 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Euclidean Algorithm == {{:Euclidean Algorithm/Examples/341 and 527}} === Proof == <onlyinclude> {{begin-eqn}} {{eqn | n = 1 | l = 527 | r = 1 \times 341 + 186 }} {{eqn | n = 2 | l = 341 | r = 1 \times 186 + 155 }} {{eqn | n = 3 | l = 186 | r = 1 \times 155 + 31 }} {{eqn | n = | l = 155 | r = 5 \times 31 }} {{end-eqn}} </onlyinclude> Thus: :$\gcd \set {341, 527} = 31$ {{qed}} == Sources == * {{...")
- 08:24, 25 September 2022 Euclidean Algorithm/Examples/56 and 72/Integer Combination (hist | edit) [1,069 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Euclidean Algorithm == <onlyinclude> $4$ can be expressed as an integer combination of $56$ and $72$: :$8 = 4 \times 56 - 3 \times 72$ </onlyinclude> == Proof == From Euclidean Algorithm: $56$ and $72$ we have: {{:Euclidean Algorithm/Examples/56 and 72/Proof}} and so: :$\gcd \set {56, 72} = 8$ Then we have: {{begin-eqn}} {{eqn | l = 8 | r = 56 - 3 \ti...")
- 08:20, 25 September 2022 Euclidean Algorithm/Examples/56 and 72 (hist | edit) [510 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Euclidean Algorithm == <onlyinclude> The GCD of $56$ and $72$ is found to be: :$\gcd \set {56, 72} = 8$ </onlyinclude> === Integer Combination === {{:Euclidean Algorithm/Examples/56 and 72/Integer Combination}} == Proof == {{:Euclidean Algorithm/Examples/56 and 72/Proof}} Categor...")
- 08:19, 25 September 2022 Euclidean Algorithm/Examples/56 and 72/Proof (hist | edit) [423 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Euclidean Algorithm == {{:Euclidean Algorithm/Examples/56 and 72}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | n = 1 | l = 72 | r = 1 \times 56 + 16 }} {{eqn | n = 2 | l = 56 | r = 3 \times 16 + 8 }} {{eqn | n = 3 | l = 16 | r = 2 \times 8 }} {{end-eqn}} </onlyinclude> Thus: :$\gcd \set {56, 72} = 8$ {{qed}} Category:Examples of Euclidean Algorithm")
- 07:02, 25 September 2022 Open Balls whose Distance between Centers is Twice Radius are Disjoint/Proof 2 (hist | edit) [790 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Open Balls whose Distance between Centers is Twice Radius are Disjoint}} == Proof == <onlyinclude> Let $z \in \map {B_r} x$. Then: {{begin-eqn}} {{eqn | l = \map d {z, y} | o = \ge | r = \size{ \map d {z, x} - \map d {y, x} } | c = Reverse Triangle Inequality }} {{eqn | o = \ge | r = \map d {y, x} - \map d {z, x} | c = {{Defof|Absolute Value}} }} {{eqn | r = \map d {x, y} - \map d {z, x} | c = {{Metric-space-axio...")
- 07:02, 25 September 2022 Open Balls whose Distance between Centers is Twice Radius are Disjoint/Proof 1 (hist | edit) [1,059 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Open Balls whose Distance between Centers is Twice Radius are Disjoint}} == Proof == <onlyinclude> {{AimForCont}} $\map {B_r} x \cap \map {B_r} y \ne \O$. Then: {{begin-eqn}} {{eqn | q = \exists z \in A | l = z \in \map {B_r} x | o = \text { and } | r = z \in \map {B_r} y | c = {{Defof|Set Intersection}} }} {{eqn | ll= \leadsto | l = \map d {x, z} < r | o = \text { and } | r = \map d {z, y} < r | c = {{D...")
- 22:46, 24 September 2022 Parameterization of Unit Circle is Simple Loop (hist | edit) [3,569 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $\mathbb S^1$ denote the unit circle whose center is at the origin of the Euclidean space $\R^2$. Let $p: \closedint 0 1 \to \R^2$ be defined by: :$\forall t \in \closedint 0 1 : \map p t = \tuple {\map \cos {2 \pi t}, \map \sin {2 \pi t} }$ Then $p$ is a simple loop with Definition:Image of Ma...")
- 17:53, 24 September 2022 Inner Product/Examples/Sequences with Finite Support (hist | edit) [4,007 bytes] Lord Farin (talk | contribs) (Created page with "== Example of Inner Product == <onlyinclude> Let $\GF$ be a subfield of $\C$. Let $V$ be the vector space of sequences with finite support over $\GF$. Let $f: \N \to \GF$ be a mapping. Let $\innerprod \cdot \cdot: V \times V \to \GF$ be the mapping defined by: :$\ds \innerprod {\sequence {a_n} } {\sequence {b...")
- 17:50, 24 September 2022 Inner Product/Examples (hist | edit) [283 bytes] Lord Farin (talk | contribs) (Created page with "== Examples of Inner Products == <onlyinclude> === Sequences with Finite Support === {{:Inner Product/Examples/Sequences with Finite Support}}</onlyinclude> Category:Examples of Inner Products")
- 22:02, 23 September 2022 Simple Loop in Hausdorff Space is Homeomorphic to Quotient Space of Interval (hist | edit) [2,756 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $\struct { X, \tau_X }$ be a Hausdorff space. Let $\gamma : \closedint 0 1 \to X$ be a simple loop. Let $\sim$ be an equivalence relation on $\closedint 0 1$ defined by: {{begin-eqn}} {{eqn | o = | r = \forall t_1 \in \openint 0 1 , t_2 \in \closedint 0 1 : | rr = t_1 \sim t_2 \iff t_2 = t_1 }} {{eqn | o = | r = \forall t_1 \in \...")
- 20:06, 23 September 2022 Linear Transformation has Finite Index iff Pseudoinverse exists (hist | edit) [643 bytes] Usagiop (talk | contribs) (Created page with "== Definition == <onlyinclude> Let $U, V$ be vector spaces over a field $K$. Let $T: U \to V$ be a linear transformation. Then $T$ has finite index {{iff}} $T$ has a pseudoinverse. </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{Bo...")
- 15:17, 23 September 2022 Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 2 (hist | edit) [2,309 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces. Let $g: S_1 \to S_2$ be a continuous surjection. Let $\RR_g \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $g$: :$\tuple {s_1, s_2} \in \RR_g \iff \map g {s_1} = \map g {s_2}$ Let $q_{\RR_g}: S_1 \to S_1 / \RR_g$...")
- 14:57, 23 September 2022 Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 1 (hist | edit) [2,402 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $g: S_1 \to S_2$ be a continuous surjection. Let $\RR_g \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $g$: :$\tuple {s_1, s_2} \in \RR_g \iff \map g {s_1} = \map g {s_2}$ Let $q_{\RR_g}: S_1 \to...")
- 10:02, 23 September 2022 Semi-Inner Product/Examples/Sequences with Finite Support (hist | edit) [3,735 bytes] Lord Farin (talk | contribs) (Created page with "== Example of Semi-Inner Product == <onlyinclude> Let $\F$ be a subfield of $\C$. Let $V$ be the vector space of sequences with finite support over $\F$. Let $\innerprod \cdot \cdot: V \times V \to \F$ be the mapping defined by: :$\ds \innerprod {\sequence{a_n} } {\sequence{b_n} } = \sum_{n = 1}^\infty a_{2n} \overline{ b_{2n}...")
- 09:53, 23 September 2022 Semi-Inner Product/Examples (hist | edit) [308 bytes] Lord Farin (talk | contribs) (Created page with "== Examples of Semi-Inner Products == <onlyinclude> {{WIP}}</onlyinclude> Category:Examples of Semi-Inner Products")
- 09:36, 23 September 2022 Expression for bilinear function (hist | edit) [7,104 bytes] Ivar Sand (talk | contribs) (Created page with "== Theorem == Let $f$ be a real function of two independent variables, $f \in \R \times \R \to \R$. Then: :$\map f {x, y}$ is a linear function of $x$ when $y$ is equal to a real constant :$\map f {x, y}$ is a linear function of $y$ when $x$ is equal to a Defini...")
- 07:53, 23 September 2022 Vector Space of Sequences with Finite Support is Vector Space (hist | edit) [1,247 bytes] Lord Farin (talk | contribs) (Created page with "== Theorem == Let $\struct {K, +, \circ}$ be a division ring. Let $V$ be the vector space of sequences with finite support in $K$. Then $V$ is a vector space over $K$. == Proof == {{WIP}} Category:Examples of Vector Spaces")
- 07:07, 23 September 2022 Euclidean Algorithm/Examples/272 and 1479 (hist | edit) [906 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Euclidean Algorithm == <onlyinclude> The GCD of $272$ and $1479$ is: :$\gcd \set {272, 1479} = 9$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | n = 1 | l = 1479 | r = 5 \times 272 + 119 }} {{eqn | n = 2 | l = 272 | r = 2 \times 119 + 34 }} {{eqn | n = 3 | l = 119 | r = 3 \times 34 + 17 }} {{eqn | n = 4 | l = 34 | r = 2 \times 17 }} {{end-eqn}...")
- 07:02, 23 September 2022 Euclidean Algorithm/Examples/306 and 657 (hist | edit) [878 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Euclidean Algorithm == <onlyinclude> The GCD of $306$ and $657$ is: :$\gcd \set {306, 657} = 9$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | n = 1 | l = 657 | r = 2 \times 306 + 45 }} {{eqn | n = 2 | l = 306 | r = 6 \times 45 + 36 }} {{eqn | n = 3 | l = 45 | r = 1 \times 36 + 9 }} {{eqn | n = 4 | l = 36 | r = 4 \times 9 }} {{end-eqn}} Thus:...")
- 06:41, 23 September 2022 Euclidean Algorithm/Examples/143 and 227 (hist | edit) [1,138 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Euclidean Algorithm == <onlyinclude> The GCD of $143$ and $227$ is: :$\gcd \set {143, 227} = 1$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | n = 1 | l = 227 | r = 1 \times 143 + 84 }} {{eqn | n = 2 | l = 143 | r = 1 \times 84 + 59 }} {{eqn | n = 3 | l = 84 | r = 1 \times 59 + 25 }} {{eqn | n = 4 | l = 59 | r = 2 \times 25 + 9 }} {{eqn | n =...")
- 06:23, 23 September 2022 Greatest Common Divisor of Set of Integers/Examples/49, 210, 350 (hist | edit) [851 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Greatest Common Divisor of Set of Integers == <onlyinclude> Let $S = \set {49, 210, 350}$. Their greatest common divisor is: :$\map \gcd S = 7$ </onlyinclude> == Proof == {{ProofWanted|To do this formally, we need to first prove that $\gcd \set {a, b, c} {{=}} \gcd \set {\gcd \set {a, b}, c}$ or something similar}} == Sources == * {{BookR...")
- 06:23, 23 September 2022 Greatest Common Divisor of Set of Integers/Examples/39, 42, 54 (hist | edit) [851 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Greatest Common Divisor of Set of Integers == <onlyinclude> Let $S = \set {39, 42, 54}$. Their greatest common divisor is: :$\map \gcd S = 3$ </onlyinclude> == Proof == {{ProofWanted|To do this formally, we need to first prove that $\gcd \set {a, b, c} {{=}} \gcd \set {\gcd \set {a, b}, c}$ or something similar == Sources == * {{BookRefer...")
- 06:15, 23 September 2022 Greatest Common Divisor of Set of Integers/Examples (hist | edit) [544 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Greatest Common Divisors of Sets of Integers == <onlyinclude> === Example: $39$, $42$, $54$ === {{:Greatest Common Divisor of Set of Integers/Examples/39, 42, 54}} === Example: $49$, $210$, $350$ === {{:Greatest Common Divisor of Set of Integers/Examples/49, 210, 3...")
- 06:00, 23 September 2022 Definition:Greatest Common Divisor of Set of Integers/Definition 2 (hist | edit) [1,363 bytes] Prime.mover (talk | contribs) (Created page with "== Definition == Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero). <onlyinclude> The '''greatest common divisor''' of $S$: :$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$ is defined as the (strictly) positive integer $d \in \Z_{>0}$ such that: {{begin-axiom}} {{axiom | q = \forall...") originally created as "Greatest Common Divisor of Set of Integers/Definition 2"
- 05:55, 23 September 2022 Definition:Greatest Common Divisor of Set of Integers/Definition 1 (hist | edit) [1,498 bytes] Prime.mover (talk | contribs) (Created page with "== Definition == <onlyinclude> Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero). <onlyinclude> The '''greatest common divisor''' of $S$: :$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$ is defined as the largest $d \in \Z_{>0}$ such that: :$\forall x \in S: d \divides x$ where $\divides$ denotes Definition:Divisor of Integ...") originally created as "Greatest Common Divisor of Set of Integers/Definition 1"
- 22:28, 22 September 2022 Continuous Surjection Induces Continuous Bijection from Quotient Space (hist | edit) [3,151 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $g: S_1 \to S_2$ be a continuous surjective mapping. Let $\RR_g \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $g$: :$\tuple {s_1, s_2} \in \RR_g \iff \map g {s_1} = \map g {s_2}$ Let $q_{\RR_g}:...")
- 20:45, 22 September 2022 Semi-Inner Product with Zero Vector (hist | edit) [1,328 bytes] Lord Farin (talk | contribs) (Created page with "== Theorem == Let $\struct{ V, \innerprod \cdot \cdot }$ be a semi-inner product space. Let $\mathbf 0_V$ be the zero vector of $V$. Then for all $v \in V$: :$\innerprod {\mathbf 0_V} v = \innerprod v {\mathbf 0_V} = 0$ == Proof == {{begin-eqn}} {{eqn| l = \innerprod {\mathbf 0_V} v | r = \innerprod {0 \cdot \mathbf 0_V} v }} {{eqn| r = 0 \cdot \innerprod {\mathbf 0_V} v | c = Definiti...")
- 18:17, 22 September 2022 Size of Tree is One Less than Order/Necessary Condition/Induction Step (hist | edit) [1,089 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let the following hold: :For all $j \le k$, a tree of order $j$ is of size $j - 1$. Then this holds: :A tree of order $k + 1$ is of size $k$. </onlyinclude> == Proof 1 == {...")
- 22:37, 21 September 2022 Injective Quotient Mapping Equals Homeomorphism (hist | edit) [2,662 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping. Then $f$ is an injective quotient mapping, {{iff}} $f$ is a homeomorphism. == Proof == === Sufficient condition === Suppose $f$ is an Definition:Inject...")
- 21:49, 21 September 2022 Quotient Mapping and Continuous Mapping Induces Continuous Mapping/Corollary (hist | edit) [2,856 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$, $T_3 = \struct {S_3, \tau_3}$ be topological spaces. Let $p: S_1 \to S_2$ be a quotient mapping. Let $g: S_2 \to S_3$ be a mapping such that for all $s_1 , s_2 \in S_1$ with $\map p {s_1} = \map p {s_2}$, we have $\map g {s_1} = \map g {s_2}$. Then $g$ induces a Definition:Mapping|mapping...")
- 16:53, 21 September 2022 Set of Integers with GCD of 1 are not necessarily Pairwise Coprime (hist | edit) [1,308 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $S$ be a set of integers such that $S$ has more than $2$ elements: :$S = \set {s_1, s_2, \ldots, s_n}$ Let: :$\map \gcd S = 1$ where $\gcd$ denotes the GCD of $S$. Then it is not necessarily the case that: :$\exists i, j \in \set {1, 2, \ldots, n}: \gcd \set {s_i, s_j} = 1$ == Proof == Proof by Counterexample Let $S = \...")
- 15:23, 21 September 2022 Quotient Mapping and Continuous Mapping Induces Continuous Mapping (hist | edit) [2,723 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$, $T_3 = \struct {S_3, \tau_3}$ be topological spaces. Let $p: S_1 \to S_2$ be a quotient mapping. Let $g: S_2 \to S_3$ be a mapping such that for all $s_1 , s_2 \in S_1$ with $\map p {s_1} = \map p {s_2}$, we have $\map g {s_1} = \map g {s_2}$. Then $g$ induces a Definition:Mapping|mapping...")
- 22:53, 20 September 2022 Composite of Quotient Mappings in Topology is Quotient Mapping (hist | edit) [1,708 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$, $T_3 = \struct {S_3, \tau_3}$ be topological spaces. Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be quotient mappings. Then $g \circ f : S_1 \to S_3$ is a quotient mapping. == Proof == Composite of Surjections is Surjection shows that $g \circ f$ is Definition:Surj...")
- 22:33, 20 September 2022 Continuous Open Surjective Mapping is Quotient Mapping (hist | edit) [1,234 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a continuous open surjective mapping. Then $f$ is a quotient mapping. == Proof == Let $U \subseteq S_2$ such that $f^{-1} \sqbrk U$ is Definition:Open Set (Topolo...")
- 21:59, 20 September 2022 Lowest Common Multiple of Integers/Examples/3054 and 12378 (hist | edit) [1,195 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Lowest Common Multiple of Integers == <onlyinclude> The lowest common multiple of $3054$ and $12378$ is: :$\lcm \set {3054, 12378} = 6 \, 300\, 402$ </onlyinclude> == Proof == From Euclidean Algorithm: $12378$ and $3054$: :$\gcd \set {3054, 12378} = 6$ Then: {{begin-eqn}} {{eqn | l = \lcm \set {3054, 12378}...")
- 12:53, 20 September 2022 Dilation of Convex Set in Vector Space is Convex (hist | edit) [1,211 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\Bbb F \in \set {\R, \C}$. Let $X$ be a vector space over $\Bbb F$. Let $C \subseteq X$ be a convex subset of $X$. Let $\alpha \in \C$. Then $\alpha C$ is convex. </onlyinclude> == Proof == Consider first the case $\alpha = 0$. We then have $\alpha C = \set { {\mathbf 0}_X}$. This is Definition:Convex Set (Vector Spa...")
- 06:25, 20 September 2022 Product of GCD and LCM/Proof 5 (hist | edit) [1,519 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Product of GCD and LCM}} == Proof == <onlyinclude> Let $d := \gcd \set {a, b}$. Then by definition of the GCD, there exist $r, s\in \Z$ such that $a = d r$ and $b = d s$. Let $m = \dfrac {a b} d$. Then: :$m = a s = r b$ which makes $m$ a common multiple of $a$ and $b$. Let $c \in \Z_{>0}$ be a common multiple of $a$ and $b$. Let us sa...")
- 06:25, 20 September 2022 Upper Bound for Lowest Common Multiple (hist | edit) [1,020 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $a, b \in \Z$ be integers such that $a b \ne 0$. Then: :$\lcm \set {a, b} \le \size {a b}$ where: :$\lcm \set {a, b}$ denotes the lowest common multiple of $a$ and $b$ == Proof == By Product of GCD and LCM: :$\lcm \set {a, b} \times \gcd \set {a, b} = \size {a b}$ where: :$\gcd \set {a, b}$ denotes the Definition:Greatest Common Divisor of Integers|greatest common diviso...")
- 23:13, 19 September 2022 Continuous Closed Surjective Mapping is Quotient Mapping (hist | edit) [2,001 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a continuous closed surjective mapping. Then $f$ is a quotient mapping. == Proof == Let $U \subseteq S_2$ such that $f^{-1} \sqbrk U$ is Definition:Open Set (To...")
- 20:25, 19 September 2022 Quotient Mapping Induces Homeomorphism between Quotient Space and Image (hist | edit) [2,512 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a quotient mapping. Let $\RR_f \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $f$: :$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$ Let $q_{\RR_f}: S_1 \to S_1 / \RR_f$ be the Definition:...")
- 22:41, 18 September 2022 Quotient Topology is Topology (hist | edit) [1,041 bytes] Anghel (talk | contribs) (This will hopefully do as proof.)
- 22:27, 18 September 2022 Quotient Mapping equals Surjective Identification Mapping (hist | edit) [2,303 bytes] Anghel (talk | contribs) (Created page with "== Theorem == Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping. Then $f$ is a quotient mapping, {{Iff}}: :$f$ is surjective, and $\tau_2$ is the identification topology on $S_2$ with respect to $f$ and $T_1$. == Proof == === Su...")
- 12:03, 18 September 2022 Volume of Paraboloid (hist | edit) [1,666 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The volume of paraboloid is half the volume of its circumscribing cylinder. == Proof == {{ProofWanted}} Category:Paraboloids")
- 11:59, 18 September 2022 Ordering of Series of Ordered Sequences/Proof 2 (hist | edit) [562 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Ordering of Series of Ordered Sequences}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 0}^\infty b_n - \sum_{n \mathop = 0}^\infty a_n | r = \sum_{n \mathop = 0}^\infty \paren {b_n - a_n} | c = Linear Combination of Convergent Series }} {{eqn | r = b_0 - a_0 + \sum_{n \mathop = 1}^\infty \paren {b_n - a_n} }} {{eqn | o = \ge | r = b_0 - a_0 | c = as $b_n - a_n > 0$ }} {{eqn | o = > | r = 0 }} {...")
- 11:59, 18 September 2022 Ordering of Series of Ordered Sequences/Proof 1 (hist | edit) [923 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Ordering of Series of Ordered Sequences}} == Proof == <onlyinclude> Let $\sequence {\epsilon_n}$ be the real sequence defined by: :$\forall n \in \N : b_n - a_n$ From Linear Combination of Convergent Series, $\ds \sum_{n \mathop = 0}^\infty \epsilon_n$ is convergent with sum $\epsilon > 0$. Then: {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 0}^\infty b_n - \sum_{n \mathop = 0}^\infty a...")
- 19:42, 17 September 2022 Path as Parameterization of Contour/Corollary 2 (hist | edit) [1,687 bytes] Anghel (talk | contribs) (Created page with "== Theorem == {{:Path as Parameterization of Contour}} If $\gamma$ is a Jordan arc, then $C$ is a simple contour, and if $\gamma$ is a Jordan curve, then $C$ is a simple closed contour. == Proof == Let $k_1, k_2 \in \set {1, \ldots, n}$, and $t_1 \in \hointr {a_{k_1 - 1} } {a_{...")
- 19:36, 17 September 2022 Path as Parameterization of Contour/Corollary 1 (hist | edit) [743 bytes] Anghel (talk | contribs) (Created page with "== Theorem == {{:Path as Parameterization of Contour}} If $\gamma$ is a closed path, then $C$ is a closed contour. == Proof == By definition of closed path, we have :$\map \gamma a = \map {\gamma_1} {a_0} = \map {\gamma_n} {a_n}$ so: :$C_1$ has start point $\map \gamma a$ and: :$C_n$ has...")
- 11:50, 17 September 2022 Existence of Lowest Common Multiple/Proof 3 (hist | edit) [1,353 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Existence of Lowest Common Multiple}} == Proof == <onlyinclude> Note that as Integer Divides Zero, both $a$ and $b$ are divisors of zero. Thus by definition $0$ is a common multiple of $a$ and $b$. Non-trivial common multiples of $a$ and $b$ exist. Indeed, $a b$ and $-\paren {a b}$ are Definition:Common Multiple|common mu...")