Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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Contraction Mapping Theorem on a Nonempty Complete Metric Space
theoremthm:contraction-mapping-complete-metric-space-2026bAnalysis Topology
Let $(X,d)$ be a complete metric space, and suppose that $X$ is nonempty. Let $T:X\to X$ be a contraction. Then $T$ has a unique fixed point in $X$ . Moreover, for every $x_0\in X$ , the iterated sequence $x_{m+1}=T(x_m)\qquad (m\in\mathbb{N}\cup\{0\})$ converges to that fixed…
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.944630Z
Contraction of a Metric Space
definitiondef:contraction-metric-space-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $T:X\to X$ be a map. We say that $T$ is a contraction if there exists a real number $\lambda$ satisfying $0\le \lambda<1$ such that $d(T(x),T(y))\le \lambda\, d(x,y)$ for every $x,y\in X$ .
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.911236Z
Fixed Point of a Self-Map
definitiondef:fixed-point-self-map-2026aAnalysis Topology
Let $X$ be a set, and let $T:X\to X$ be a map. A point $x\in X$ is called a fixed point of $T$ if $T(x)=x.$
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.878131Z
Complete Metric Space
definitiondef:complete-metric-space-2026aAnalysis Topology
Let $(X,d)$ be a metric space. We say that $(X,d)$ is complete if every Cauchy sequence in $(X,d)$ converges to a point of $X$ .
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.810854Z
Orientable and Oriented Smooth Manifold with Boundary
definitiondef:oriented-smooth-manifold-boundary-2026aGeometry Topology
Let $M$ be a smooth manifold with boundary. We say that $M$ is orientable if it admits an oriented smooth atlas in the sense of Oriented Smooth Atlas on a Smooth Manifold with Boundary. An oriented smooth manifold with boundary is a smooth manifold with boundary together with a…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.696956Z
Oriented Smooth Atlas on a Smooth Manifold with Boundary
definitiondef:oriented-smooth-atlas-manifold-boundary-2026aGeometry Topology
Let $M$ be a smooth manifold with boundary. An oriented smooth atlas on $M$ is a smooth atlas $\mathcal{A}=\bigl((U_\alpha,\varphi_\alpha)\bigr)_{\alpha\in A}$ such that for every $\alpha,\beta\in A$ , the charts $(U_\alpha,\varphi_\alpha)$ and $(U_\beta,\varphi_\beta)$ are po…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.660708Z
Positive Compatibility of Charts on the Interior of a Smooth Manifold with Boundary
definitiondef:positive-compatibility-charts-interior-manifold-boundary-2026aGeometry Topology Multivariable Calculus
Let $M$ be a smooth manifold with boundary of dimension $n$ , and let $(U,\varphi)$ and $(V,\psi)$ be charts in the chosen smooth atlas. We say that these charts are positively compatible if either $U\cap V\cap \operatorname{int}(M)=\varnothing,$ or else the following conditio…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.622615Z
Second Countable Topological Space
definitiondef:second-countable-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a topological space. We say that $X$ is second countable if there exists a basis $\mathcal{B}$ for $\mathcal{T}$ in the sense of Basis for a Topology such that $\mathcal{B}$ is countable.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.298035Z
Basis for a Topology
definitiondef:basis-topology-2026aTopology
Let $(X,\mathcal{T})$ be a topological space. A basis for the topology $\mathcal{T}$ is a family $\mathcal{B}\subseteq \mathcal{T}$ with the following property: for every point $x\in X$ and every open set $U\in\mathcal{T}$ satisfying $x\in U$ , there exists a set…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.256697Z
Hausdorff Topological Space
definitiondef:hausdorff-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a topological space. We say that $X$ is Hausdorff if for every two distinct points $x,y\in X$ there exist open sets $U,V\in\mathcal{T}$ such that $x\in U,\qquad y\in V,\qquad U\cap V = \varnothing.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.212310Z
Interior Points, Boundary Points, and Boundary of a Smooth Manifold with Boundary
definitiondef:boundary-smooth-manifold-with-boundary-2026aGeometry Topology Multivariable Calculus
Let $M$ be a smooth manifold with boundary of dimension $n$ in the sense of Smooth Atlas and Smooth Manifold with Boundary, and let $p\in M$ . We say that $p$ is an interior point of $M$ if there exists a chart $(U,\varphi)$ in the chosen atlas with $p\in U$ and…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.620708Z
Smooth Atlas and Smooth Manifold with Boundary
definitiondef:smooth-manifold-with-boundary-2026aGeometry Topology Multivariable Calculus
Let $(X,\mathcal{T})$ be a topological space, and let $n\in\mathbb{N}$ . A smooth atlas of dimension $n$ on $X$ , modeled on the closed upper half-space, is a family of charts $\mathcal{A}=\bigl((U_\alpha,\varphi_\alpha)\bigr)_{\alpha\in A}$ for some set $A$ such that the follo…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.568526Z
Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space
definitiondef:smooth-compatible-charts-upper-half-space-2026aGeometry Topology Multivariable Calculus
Let $(X,\mathcal{T})$ be a topological space, let $n\in\mathbb{N}$ , and let $(U,\varphi)$ and $(V,\psi)$ be charts of dimension $n$ on $X$ in the sense of Chart Modeled on the Closed Upper Half-Space. Write…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.517394Z
Chart Modeled on the Closed Upper Half-Space
definitiondef:chart-upper-half-space-2026aGeometry Topology Multivariable Calculus
Let $(X,\mathcal{T})$ be a topological space, let $n\in\mathbb{N}$ , and let $U\subseteq X$ . A chart of dimension $n$ on $X$ , modeled on the closed upper half-space, is a pair $(U,\varphi)$ with the following properties. 1. $U\in\mathcal{T}$ . 2. If $H^n$ denotes the half-space fr…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.439943Z
Closed Upper Half-Space in Euclidean Space
definitiondef:closed-upper-half-space-euclidean-2026aGeometry Topology Multivariable Calculus
Let $n\in\mathbb{N}$ . In the Euclidean space $\mathbb{R}^n$ , the closed upper half-space is the subset $H^n=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n : x_n\ge 0\}.$ A subset $\Omega\subseteq H^n$ is said to be open in $H^n$ if there exists an open subset $U\subseteq\mathbb{R}^n$ suc…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.308760Z
Open Ball in a Metric Space is Open
theoremthm:open-ball-metric-space-open-2026aTopology
Let $(X,d)$ be a metric space, let $x\in X$ , and let $r\in\mathbb{R}$ satisfy $r>0$ . Then the open ball $B_d(x,r)$ is open in the metric space $(X,d)$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:53:48.537806Z
Heine-Borel Theorem in $\mathbb{R}^n$
theoremthm:heine-borel-rn-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A\subseteq\mathbb{R}^n$ . Then the following are equivalent. 1. $A$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the Euclidean distance. 2. $A$ is closed in $\mathbb{R}^n$ a…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.209722Z
Compact Subset of $\mathbb{R}^n$ is Closed
theoremthm:compact-subset-rn-closed-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A\subseteq\mathbb{R}^n$ . Assume that $A$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the Euclidean distance. Then $A$ is closed in $\mathbb{R}^n$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.154631Z
Compact Subset of $\mathbb{R}^n$ is Bounded
theoremthm:compact-subset-rn-bounded-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A\subseteq\mathbb{R}^n$ . Assume that $A$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the Euclidean distance. Then $A$ is bounded as a subset of the metric space…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.110472Z
Closed Box in $\mathbb{R}^n$ is Compact
theoremthm:closed-box-compact-rn-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ . For each index $i\in\{1,\dots,n\}$ , let $a_i,b_i\in\mathbb{R}$ satisfy $a_i\le b_i$ , and let $B\subseteq\mathbb{R}^n$ be the closed box determined by these endpoints. Then $B$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological sp…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.055462Z

Showing 81-100 of 124

Back Next

Contraction Mapping Theorem on a Nonempty Complete Metric Space

Contraction of a Metric Space

Fixed Point of a Self-Map

Complete Metric Space

Orientable and Oriented Smooth Manifold with Boundary

Oriented Smooth Atlas on a Smooth Manifold with Boundary

Positive Compatibility of Charts on the Interior of a Smooth Manifold with Boundary

Second Countable Topological Space

Basis for a Topology

Hausdorff Topological Space

Interior Points, Boundary Points, and Boundary of a Smooth Manifold with Boundary

Smooth Atlas and Smooth Manifold with Boundary

Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space

Chart Modeled on the Closed Upper Half-Space

Closed Upper Half-Space in Euclidean Space

Open Ball in a Metric Space is Open

Heine-Borel Theorem in Rn\mathbb{R}^nRn

Compact Subset of Rn\mathbb{R}^nRn is Closed

Compact Subset of Rn\mathbb{R}^nRn is Bounded

Closed Box in Rn\mathbb{R}^nRn is Compact

Heine-Borel Theorem in $\mathbb{R}^n$

Compact Subset of $\mathbb{R}^n$ is Closed

Compact Subset of $\mathbb{R}^n$ is Bounded

Closed Box in $\mathbb{R}^n$ is Compact