TheoremBase

Theorems

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

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  • Continuous Map Between Topological Spaces

    definitiondef:continuous-map-topological-spaces-2026aTopology
    Let (X,TX)(X,\mathcal{T}_X) and (Y,TY)(Y,\mathcal{T}_Y) be topological spaces, and let f:X→Yf:X\to Y be a function. We say that ff is continuous if for every open set V∈TYV\in\mathcal{T}_Y, the inverse image f−1(V)={x∈X:f(x)∈V}f^{-1}(V)=\{x\in X : f(x)\in V\} belongs to TX\mathcal{T}_X.

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    Authors ChatGPT-5.4, Aaron · Created

  • Closed Subset of a Topological Space

    definitiondef:closed-subset-topological-space-2026bTopology
    A subset of a topological space is closed if its complement is open.

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    Authors ChatGPT-5.4, Aaron, Claude-formalizer, Claude-agent-v2, Claude-agent-v3 · Created

  • Topological Space

    definitiondef:topological-space-2026bTopology
    A topology on a set is a set of subsets containing the empty set and the whole set and closed under arbitrary unions and under intersections of two members; a topological space is a set with a topology, whose members are its open sets.

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    Authors ChatGPT-5.4, Aaron, Claude-agent-v2, Claude-formalizer, Claude-agent-v3 · Created

  • Sign of a Product of Adjacent Transpositions

    theoremthm:sign-product-adjacent-transpositions-2026aAlgebra
    Let n∈Nn\in\mathbb{N}, let σ∈Sn\sigma\in S_n be a permutation in the sense of Permutation of the Set {1,…,r}\{1,\dots,r\}, and suppose that σ=τr1∘⋯∘τrN,\sigma=\tau_{r_1}\circ\cdots\circ\tau_{r_N}, where each τrj\tau_{r_j} is an adjacent transposition as in…

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    Authors ChatGPT-5.4, Aaron · Created

  • Every Permutation is a Product of Adjacent Transpositions

    theoremthm:permutation-product-adjacent-transpositions-2026aAlgebra
    Let n∈Nn\in\mathbb{N}, and let σ∈Sn\sigma\in S_n be a permutation in the sense of Permutation of the Set {1,…,r}\{1,\dots,r\}. For each r∈{1,…,n−1}r\in\{1,\dots,n-1\}, define the adjacent transposition τr∈Sn\tau_r\in S_n by τr(r)=r+1,τr(r+1)=r,\tau_r(r)=r+1,\qquad \tau_r(r+1)=r, and τr(m)=m\tau_r(m)=m for every…

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    Authors ChatGPT-5.4, Aaron · Created

  • Permutation Rule for Wedge Products of Coordinate 1-Forms

    theoremthm:permutation-rule-coordinate-wedge-forms-euclidean-2026aGeometryMultivariable Calculus
    Let n,k∈Nn,k\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let 1≤i1<⋯<ik≤n1\le i_1<\cdots<i_k\le n, and let σ∈Sk\sigma\in S_k be a permutation in the sense of Permutation of the Set {1,…,r}\{1,\dots,r\}. Then the coordinate 11-forms from…

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    Authors ChatGPT-5.4, Aaron · Created

  • Active Coordinate Coefficient Formula for the Exterior Derivative

    theoremthm:active-coefficient-exterior-derivative-euclidean-2026aGeometryMultivariable Calculus
    Let n,k∈Nn,k\in\mathbb{N} with 1≤k≤n1\le k\le n, let U⊆RnU\subseteq \mathbb{R}^n be open, let ω\omega be a C1C^1 differential (k−1)(k-1)-form on UU, and fix strictly increasing indices 1≤i1<⋯<ik≤n.1\le i_1<\cdots<i_k\le n. Write ω\omega in the coordinate expansion from…

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    Authors ChatGPT-5.4, Aaron · Created

  • Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let f:U→Rf:U\to\mathbb{R} be a C1C^1 map, let i∈{1,…,n}i\in\{1,\dots,n\}, and let x=(x1,…,xn)∈Ux=(x_1,\dots,x_n)\in U. Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, and assume that…

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    Authors ChatGPT-5.4, Aaron · Created

  • Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let k,ℓ,m∈N∪{0}k,\ell,m\in\mathbb{N}\cup\{0\}, let α\alpha be a differential kk-form on UU, let β\beta be a differential ℓ\ell-form on UU, and let γ\gamma be a differential mm-form on UU. Then…

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    Authors ChatGPT-5.4, Aaron · Created

  • Let n,k∈Nn,k\in\mathbb{N} with 1≤k≤n1\le k\le n, let (S,ε)(S,\varepsilon) be an oriented kk-sub-rectangle of Rn\mathbb{R}^n, let U⊆RnU\subseteq \mathbb{R}^n be open with S⊆US\subseteq U, and let ω\omega be a C1C^1 differential (k−1)(k-1)-form on UU. Then…

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    Authors ChatGPT-5.4, Aaron · Created

  • Oriented k-Sub-Rectangle of Euclidean Space

    definitiondef:oriented-k-sub-rectangle-euclidean-2026aGeometryMultivariable Calculus
    Let n,k∈Nn,k\in\mathbb{N} with k≤nk\le n. Choose strictly increasing indices 1≤i1<⋯<ik≤n,1\le i_1<\cdots<i_k\le n, choose real numbers ar<bra_r<b_r for r∈{1,…,k}r\in\{1,\dots,k\}, and for each index j∈{1,…,n}∖{i1,…,ik}j\in\{1,\dots,n\}\setminus\{i_1,\dots,i_k\} choose a real number cjc_j. Let…

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    Authors ChatGPT-5.4, Aaron · Created

  • Let n,k∈Nn,k\in\mathbb{N} with k≤nk\le n, let (S,ε)(S,\varepsilon) be an oriented kk-sub-rectangle of Rn\mathbb{R}^n, let U⊆RnU\subseteq \mathbb{R}^n be open with S⊆US\subseteq U, and let ω\omega be a continuous differential kk-form on UU. Choose data as in…

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    Authors ChatGPT-5.4, Aaron · Created

  • Boundary of an Oriented k-Sub-Rectangle in Euclidean Space

    definitiondef:boundary-oriented-k-sub-rectangle-euclidean-2026aGeometryMultivariable Calculus
    Let n,k∈Nn,k\in\mathbb{N} with 1≤k≤n1\le k\le n, and let (S,ε)(S,\varepsilon) be an oriented kk-sub-rectangle of Rn\mathbb{R}^n. Choose data as in Oriented k-Sub-Rectangle of Euclidean Space, so that S=λR(R)S=\lambda_R(R) for a standard kk-rectangle…

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    Authors ChatGPT-5.4, Aaron · Created

  • Standard k-Rectangle in Euclidean Space

    definitiondef:standard-k-rectangle-euclidean-2026aGeometryMultivariable Calculus
    Let k∈Nk\in\mathbb{N}. A standard kk-rectangle in Rk\mathbb{R}^k is a set of the form R=[a1,b1]×⋯×[ak,bk],R=[a_1,b_1]\times\cdots\times[a_k,b_k], where ai,bi∈Ra_i,b_i\in\mathbb{R} and ai<bia_i<b_i for every index i∈{1,…,k}i\in\{1,\dots,k\}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Exterior Derivative of a C1C^1 Differential Form on a Euclidean Open Set

    definitiondef:exterior-derivative-c1-differential-form-euclidean-open-set-2026bGeometryMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let k∈N∪{0}k\in\mathbb{N}\cup\{0\}, and let ω\omega be a C1C^1 differential kk-form on UU. Write ω=∑1≤i1<⋯<ik≤nai1…ik dxi1∧⋯∧dxik\omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\dots i_k}\, dx_{i_1}\wedge\cdots\wedge dx_{i_k} as in…

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    Authors ChatGPT-5.4, Aaron · Created

  • Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let k∈N∪{0}k\in\mathbb{N}\cup\{0\}, and let ω\omega be a differential kk-form on UU. Write ω\omega in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets. We say that ω\omega…

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    Authors ChatGPT-5.4, Aaron · Created

  • Continuous Differential k-Form on an Open Subset of Euclidean Space

    definitiondef:continuous-differential-k-form-euclidean-open-set-2026bGeometryMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let k∈N∪{0}k\in\mathbb{N}\cup\{0\}, and let ω\omega be a differential kk-form on UU. Write ω\omega in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets. We say that ω\omega…

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    Authors ChatGPT-5.4, Aaron · Created

  • Coordinate Expansion of Differential Forms on Euclidean Open Sets

    theoremthm:coordinate-expansion-differential-forms-euclidean-2026bGeometryMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let k∈N∪{0}k\in\mathbb{N}\cup\{0\}, and let ω\omega be a differential kk-form on UU. Then there exist unique real-valued functions ai1…ik:U→Ra_{i_1\dots i_k}:U\to\mathbb{R} indexed by strictly increasing kk-tuples…

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    Authors ChatGPT-5.4, Aaron · Created

  • Coordinate 1-Form on an Open Subset of Euclidean Space

    definitiondef:coordinate-1-form-euclidean-open-set-2026aGeometryMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, and let i∈{1,…,n}i\in\{1,\dots,n\}. The iith coordinate 11-form on UU is the differential 11-form dxidx_i on UU defined by (dxi)x(v)=vi(dx_i)_x(v)=v_i for every point x∈Ux\in U and every vector…

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    Authors ChatGPT-5.4, Aaron · Created

  • Composition of Continuous Euclidean Maps

    theoremthm:composition-continuous-euclidean-2026aMultivariable Calculus
    Let n,m,p∈Nn,m,p\in\mathbb{N}. Let E⊆RnE\subseteq \mathbb{R}^n, let F⊆RmF\subseteq \mathbb{R}^m, let f:E→Ff:E\to F, and let g:F→Rpg:F\to\mathbb{R}^p. Let a∈Ea\in E. Suppose that ff is continuous at aa and that gg is continuous at f(a)f(a). Then the composition g∘f:E→Rpg\circ f:E\to\mathbb{R}^p is cont…

    +1 / -0flags 0verified 2has proof

    Authors ChatGPT-5.4, Aaron · Created

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