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• 20:36, 2 April 2025 Austrodata talk contribs created page Definition:Local Diffeomorphism/Smooth Manifold (Created page with "== Definition == <onlyinclude> Let $S$ and $T$ be smooth manifolds with or without boundary. Let $f: S \to T$ be a mapping. Then $f$ is a '''local diffeomorphism''' {{iff}} every point $p \in S$ has a neighborhood $U$ such that: * $f \sqbrk U$ is [[Definition:Open Set|open]...")
• 16:05, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Algebra of Germs of Smooth Functions (will need to be redone/integrated with the vector space one)
• 15:53, 2 April 2025 Austrodata talk contribs created page Definition:Topological Manifold with Boundary/Coordinate Half-Ball (Created page with "== Definition == Let $M$ be a $n$-dimensional topological manifold with boundary. <onlyinclude> Let $p \in M$. Let $r \in \R_{>0}$ be a strictly positive real number. Let $\map {B_r} p$ denote an open ball of radius $r$ centered on $p$. A '''Definition:Topological Manif...")
• 15:10, 2 April 2025 Austrodata talk contribs created page Topological Manifold with Boundary is Topological Manifold iff Boundary is Empty (kinda obvious but ...)
• 13:59, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Boundary (content was: "#REDIRECT Definition:Topological Manifold with Boundary/Chart/Boundary", and the only contributor was "Austrodata" (talk))
• 13:58, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Interior (content was: "#REDIRECT Definition:Topological Manifold with Boundary/Chart/Interior", and the only contributor was "Austrodata" (talk))
• 13:58, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Interior (content was: "#REDIRECT Definition:Topological Manifold with Boundary/Interior", and the only contributor was "Austrodata" (talk))
• 13:58, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Interior Point (content was: "#REDIRECT Definition:Topological Manifold with Boundary/Interior Point", and the only contributor was "Austrodata" (talk))
• 13:58, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Chart (content was: "#REDIRECT Definition:Topological Manifold with Boundary/Chart", and the only contributor was "Austrodata" (talk))
• 13:57, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Boundary Point (content was: "#REDIRECT Definition:Topological Manifold with Boundary/Boundary Point", and the only contributor was "Austrodata" (talk))
• 13:57, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Boundary (content was: "#REDIRECT Definition:Topological Manifold with Boundary/Boundary", and the only contributor was "Austrodata" (talk))
• 13:57, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary (content was: "#REDIRECT Definition:Topological Manifold with Boundary", and the only contributor was "Austrodata" (talk))
• 13:56, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Closed Upper Half-Space/Boundary (content was: "#REDIRECT Definition:Closed Upper Half-Space/Boundary", and the only contributor was "Austrodata" (talk))
• 13:54, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Closed Upper Half-Space (content was: "#REDIRECT Definition:Closed Upper Half-Space", and the only contributor was "Austrodata" (talk))
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary to Definition:Topological Manifold with Boundary
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary/Boundary to Definition:Topological Manifold with Boundary/Boundary
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary/Boundary Point to Definition:Topological Manifold with Boundary/Boundary Point
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary/Chart to Definition:Topological Manifold with Boundary/Chart
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Boundary to Definition:Topological Manifold with Boundary/Chart/Boundary
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Interior to Definition:Topological Manifold with Boundary/Chart/Interior
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary/Interior to Definition:Topological Manifold with Boundary/Interior
• 13:45, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Topological Manifold with Boundary/Interior Point to Definition:Topological Manifold with Boundary/Interior Point
• 13:44, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Closed Upper Half-Space/Interior (content was: "#REDIRECT Definition:Closed Upper Half-Space/Interior", and the only contributor was "Austrodata" (talk))
• 13:41, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Closed Upper Half-Space to Definition:Closed Upper Half-Space
• 13:41, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Closed Upper Half-Space/Boundary to Definition:Closed Upper Half-Space/Boundary
• 13:41, 2 April 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Closed Upper Half-Space/Interior to Definition:Closed Upper Half-Space/Interior
• 13:03, 2 April 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Topological Manifold with Boundary/Interiors and Boundaries (content was: "Let $M$ be a $n$-dimensional topological manifold with boundary. === Interior Points === {{:User:Austrodata/Definition:Topological Manifold with Boundary/Interior Point}} === User:Austrodata/Definition:Topological Mani...", and the only contributor was "Austrodata" (talk))
• 12:59, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Interiors and Boundaries (some transclusions, not very this one is OK)
• 12:45, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Boundary (Created page with "== Definition == Let $M$ be a $n$-dimensional topological manifold with boundary. <onlyinclude> The '''boundary''' of $M$, denoted $\partial M$, is the set of all its boundary points. </onlyinclude> == Sources == * {{BookReferenc...")
• 12:45, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Interior (Created page with "== Definition == Let $M$ be a $n$-dimensional topological manifold with boundary. <onlyinclude> The '''interior''' of $M$, denoted $\Int M$, is the set of all its interior point. </onlyinclude> == Sources == * {{BookReference|Int...")
• 12:44, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Boundary Point (Created page with "== Definition == Let $M$ be a $n$-dimensional topological manifold with boundary. <onlyinclude> Let $\partial \H^n$ denote the boundary of the closed upper half-space. A point $p \in M$ is an User:Austrodata/Definition:Topological Manifold with Boundary/Boundary P...")
• 12:43, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Interior Point (Created page with "== Definition == Let $M$ be a $n$-dimensional topological manifold with boundary. <onlyinclude> A point $p \in M$ is an interior point of $M$ if $p$ is in the domain of some User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Interior|inte...")
• 12:36, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Boundary (Created page with "== Definition == Let $M$ be a $n$-dimensional topological manifold with boundary. Let $\struct {U, \varphi}$ be a chart of $M$. Let $\partial \H^n$ denote the boundary of the closed upper half-space. <onlyinclude>...")
• 12:35, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Interior (Created page with "== Definition == Let $M$ be a $n$-dimensional topological manifold with boundary. Let $\struct {U, \varphi}$ be a chart of $M$. <onlyinclude> The chart $\struct {U, \phi}$ is an '''User:Austrodata/Definition:Topological Manifold with Boundary/Chart/Interior|in...")
• 12:34, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary/Chart (Created page with "== Definition == <onlyinclude> Let $M$ be a $n$-dimensional topological manifold with boundary. Let $\H^n$ denote the closed upper half-space. A '''chart''' of $M$ is an ordered pair $\struct {U, \varphi}$, where: :$U$ is an open subset of $M$...")
• 12:32, 2 April 2025 Austrodata talk contribs created page Category:Topological Manifolds with Boundary (Created page with "{{DefinitionCategory|def = Topological Manifold with Boundary|Topology}}")
• 12:24, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Topological Manifold with Boundary (Created page with "== Definition == <onlyinclude> A '''$d$-dimensional topological manifold with boundary''' is a second-countable Hausdorff space $M$ in which every point has a neighborhood homeomorphic either to: : a) an open subset of $\R^n$ or : b) an Definition:Ope...")
• 12:19, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Closed Upper Half-Space/Boundary (Created page with "== Definition == Let $\H^n$ be the closed n-dimensional upper half-space. <onlyinclude> The '''boundary of $\H^n$''', denoted $\partial \H^n$, is a subset of $\H^n$ defined as: :$\partial \H^n := \set { \tuple {x_1, \cdots, x_n} \in \R^n : x_n = 0 }$ </onlyinclude> == Sources == * {{BookReference|Introduction to Smooth Manifolds|2013|John M. Lee|ed =...")
• 12:18, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Closed Upper Half-Space/Interior (Created page with "== Definition == Let $\H^n$ be the closed n-dimensional upper half-space. <onlyinclude> The '''interior of $\H^n$''', denoted $\Int {\H^n}$, is a subset of $\H^n$ defined as: :$\Int {\H^n} := \set { \tuple {x_1, \cdots, x_n} \in \R^n : x_n > 0 }$ </onlyinclude> == Sources == * {{BookReference|Introduction to Smooth Manifolds|2013|John M. Lee|ed = 2nd...")
• 12:14, 2 April 2025 Austrodata talk contribs created page User:Austrodata/Definition:Closed Upper Half-Space (Created page with "== Definition == <onlyinclude> The '''closed n-dimensional upper half-space''', denoted $\H^n$, is a subset of the Euclidean space $\R^n$ defined as: :$\H^n := \set { \tuple {x_1, \cdots, x_n} \in \R^n : x_n \geq 0 }$ </onlyinclude> == Sources == * {{BookReference|Introduction to Smooth Manifolds|2013|John M. Lee|ed = 2nd|edpage = Second Edition}}: Chapter $1$: Smooth Manifolds: $\S$...")
• 15:32, 1 April 2025 Austrodata talk contribs deleted page Definition:Differential/Manifolds (NP. I guess I should just have adapted the links)
• 13:28, 1 April 2025 Austrodata talk contribs created page Talk:Algebra over Field/Examples/2x2 Matrices under Multiplication (complex nxn matrices -> $(2n)^2$-dimensional real algebras)
• 12:56, 1 April 2025 Austrodata talk contribs created page Definition:Space of Smooth Real Functions (Created page with "== Definition == <onlyinclude> Let $M$ be a smooth manifold with or without boundary. The set of all '''smooth real functions''' is called the '''space of smooth real functions''' and is denoted $\map {\mathrm C^\infty} {M, \R}$, often simplified to $\map {\mathrm C^\infty} M$. </onlyinclude> == Also see...")
• 11:44, 1 April 2025 Austrodata talk contribs created page Definition:Differential/Manifolds (-> [Definition:Differential of Mapping/Manifolds]) Tag: New redirect
• 11:35, 1 April 2025 Austrodata talk contribs created page Definition:Generated Subalgebra (-> [Definition:Generator of Algebra]) Tag: New redirect
• 11:27, 1 April 2025 Austrodata talk contribs created page Definition:Positive Homogeneity (-> [Definition:Homogeneous Function/Positive Homogeneity]) Tag: New redirect
• 11:13, 31 March 2025 Austrodata talk contribs deleted page User:Austrodata/Definition:Derivation of Germs of Smooth Functions (content was: "#REDIRECT Definition:Derivation of Germs of Smooth Functions", and the only contributor was "Austrodata" (talk))
• 11:13, 31 March 2025 Austrodata talk contribs moved page User:Austrodata/Definition:Derivation of Germs of Smooth Functions to Definition:Derivation of Germs of Smooth Functions (ready enough)
• 21:40, 30 March 2025 Austrodata talk contribs created page Category:Definitions/Algebra Isomorphisms (= {DefinitionCategory|def = Algebra Isomorphism|Algebra Homomorphisms|Isomorphisms (Abstract Algebra)})
• 21:30, 30 March 2025 Austrodata talk contribs created page User:Austrodata/Definition:Restriction of Functor (Created page with "Let $\mathbf A$ and $\mathbf B$ be subcategories. Let $F: \mathbf A \to \mathbf B$ be a functor. Let $\mathbf S$ be a subcategory of $\mathbf A$. The '''restriction''' of $F$ to $\mathbf S$, denoted by $F|_{\mathbf S}$, is a functor $F|_{\mathbf S}: \mathbf S \to \mathbf B$ such that: {{begin-axiom}} {{axiom | lc= Definition:Object (Category...")