Theorems

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

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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
Smooth Map on an Open Subset of Euclidean Space
definitiondef:smooth-map-euclidean-open-set-2026aMultivariable Calculus
Let $n,m\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, and let $F=(F_1,\dots,F_m):U\to\mathbb{R}^m$ . We say that $F$ is smooth on $U$ if for every multi-index $\alpha$ of length $n$ and every index $j\in\{1,\dots,m\}$ , the partial derivative of $F_j$ of order $\alpha$ exi…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.163945Z
Partial Derivative of Order $\alpha$
definitiondef:partial-derivative-order-alpha-2026aMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $f:U\to\mathbb{R}$ , and let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a multi-index of length $n$ . We define recursively what it means for the partial derivative of $f$ of order $\alpha$ to exist on $U$ , and when it…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.110781Z
Order and Factorial of a Multi-Index
definitiondef:order-factorial-multi-index-2026aCombinatorics Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a multi-index of length $n$ in the sense of Multi-Index of Length $n$ . The order of $\alpha$ is the nonnegative integer $|\alpha|=\alpha_1+\cdots+\alpha_n.$ The factorial of $\alpha$ is the natural number…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.050628Z
Multi-Index of Length $n$
definitiondef:multi-index-length-n-2026aCombinatorics Multivariable Calculus
Let $n\in\mathbb{N}$ . A multi-index of length $n$ is an element $\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}\cup\{0\})^n.$ That is, a multi-index of length $n$ is an ordered $n$ -tuple of nonnegative integers. The zero multi-index of length $n$ is $0=(0,\dots,0).$ For e…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:33.968980Z
Interior Points, Boundary Points, and Boundary of a Smooth Manifold with Boundary
definitiondef:boundary-smooth-manifold-with-boundary-2026aTopology Geometry 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-2026aTopology Geometry 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-2026aTopology Geometry 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-2026aTopology Geometry 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-2026aTopology Geometry 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
Euclidean Open Box Criterion in $\mathbb{R}^n$
theoremthm:euclidean-open-box-criterion-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ and let $U\subseteq \mathbb{R}^n$ . Then $U$ is open in the Euclidean sense if and only if for every point $x=(x_1,\dots,x_n)\in U$ there exists a real number $\delta>0$ such that every point $y=(y_1,\dots,y_n)\in\mathbb{R}^n$ satisfying…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.001092Z
Closed Interval $[a,b]$ is Compact in $\mathbb{R}$
theoremthm:closed-interval-compact-real-2026aAnalysis Topology
Let $a,b\in\mathbb{R}$ satisfy $a\le b$ . Then the interval $[a,b]$ from the interval definition is compact in $\mathbb{R}$ , where $\mathbb{R}$ is regarded as a topological space through the topology determined by the Euclidean distance.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:45.935657Z
Closed Box in $\mathbb{R}^n$
definitiondef:closed-box-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ . For each index $i\in\{1,\dots,n\}$ , let $a_i,b_i\in\mathbb{R}$ satisfy $a_i\le b_i$ . The subset $B=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n : a_i\le x_i\le b_i \text{ for every } i\in\{1,\dots,n\}\}$ is called the closed box in $\mathbb{R}^n$ determined by the…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:45.833615Z

Showing 1361-1380 of 1477

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Second Countable Topological Space

Basis for a Topology

Hausdorff Topological Space

Smooth Map on an Open Subset of Euclidean Space

Partial Derivative of Order α\alphaα

Order and Factorial of a Multi-Index

Multi-Index of Length nnn

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

Euclidean Open Box Criterion in Rn\mathbb{R}^nRn

Closed Interval [a,b][a,b][a,b] is Compact in R\mathbb{R}R

Closed Box in Rn\mathbb{R}^nRn

Partial Derivative of Order $\alpha$

Multi-Index of Length $n$

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

Euclidean Open Box Criterion in $\mathbb{R}^n$

Closed Interval $[a,b]$ is Compact in $\mathbb{R}$

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