Theorems

A growing collection of mathematical statements with user-submitted proofs.

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  • Basis for a Topology

    definitiondef:basis-topology-2026aTopology
    Let (X,T)(X,\mathcal{T}) be a \reftext{def:topological-space-2026a}{topological space}. A basis for the topology T\mathcal{T} is a family B⊆T\mathcal{B}\subseteq \mathcal{T} with the following property: for every point x∈Xx\in X and every open set U∈TU\in\mathcal{T} satisfying x∈Ux\in U, there exists a set B∈BB\in\mathcal{B} such that x∈B⊆U.x\in B\subseteq U.

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    Authors ChatGPT-5.4, Aaron · Created

  • Hausdorff Topological Space

    definitiondef:hausdorff-topological-space-2026aTopology
    Let (X,T)(X,\mathcal{T}) be a \reftext{def:topological-space-2026a}{topological space}. We say that XX is Hausdorff if for every two distinct points x,y∈Xx,y\in X there exist open sets U,V∈TU,V\in\mathcal{T} such that x∈U,y∈V,U∩V=∅.x\in U,\qquad y\in V,\qquad U\cap V = \varnothing.

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    Authors ChatGPT-5.4, Aaron · Created

  • Smooth Map on an Open Subset of Euclidean Space

    definitiondef:smooth-map-euclidean-open-set-2026aMultivariable Calculus
    Let n,m∈Nn,m\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be \reftext{def:open-subset-euclidean-space-2026a}{open}, and let F=(F1,…,Fm):U→RmF=(F_1,\dots,F_m):U\to\mathbb{R}^m. We say that FF is smooth on UU if for every multi-index α\alpha of length nn and every index j∈{1,…,m}j\in\{1,\dots,m\}, the partial derivative of FjF_j of order α\alpha exists on UU in the sense of \ref{def:partial-derivative-order-alpha-2026a}, and the resulting function ∂αFj:U→R\partial^\alpha F_j:U\to\mathbb{R} is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of UU}. A real-valued map f:U→Rf:U\to\mathbb{R} is smooth if it is smooth as a map from UU to R1\mathbb{R}^1.

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    Authors ChatGPT-5.4, Aaron · Created

  • Partial Derivative of Order α\alpha

    definitiondef:partial-derivative-order-alpha-2026aMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be \reftext{def:open-subset-euclidean-space-2026a}{open}, let f:U→Rf:U\to\mathbb{R}, and let α=(α1,…,αn)\alpha=(\alpha_1,\dots,\alpha_n) be a multi-index of length nn. We define recursively what it means for the partial derivative of ff of order α\alpha to exist on UU, and when it does exist we denote it by ∂αf.\partial^\alpha f. If α=0\alpha=0, then the derivative of order α\alpha exists on UU and is defined by ∂αf=f.\partial^\alpha f = f. Suppose α≠0\alpha\ne 0. We say that the derivative of order α\alpha exists on UU if there exists an index i∈{1,…,n}i\in\{1,\dots,n\} such that ei≤αe_i\le \alpha, the derivative ∂α−eif\partial^{\alpha-e_i}f exists on UU, and the ordinary partial derivative ∂∂xi(∂α−eif)\frac{\partial}{\partial x_i}\bigl(\partial^{\alpha-e_i}f\bigr) exists at every point of UU in the sense of \ref{def:partial-derivative-coordinate-map-2026a}. In that case we define ∂αf=∂∂xi(∂α−eif).\partial^\alpha f = \frac{\partial}{\partial x_i}\bigl(\partial^{\alpha-e_i}f\bigr). For a map F=(F1,…,Fm):U→RmF=(F_1,\dots,F_m):U\to\mathbb{R}^m, we say that the derivative of order α\alpha exists on UU if each component derivative ∂αFj\partial^\alpha F_j exists on UU, and then we define ∂αF=(∂αF1,…,∂αFm).\partial^\alpha F = (\partial^\alpha F_1,\dots,\partial^\alpha F_m).

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    Authors ChatGPT-5.4, Aaron · Created

  • Order and Factorial of a Multi-Index

    definitiondef:order-factorial-multi-index-2026aCombinatoricsMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, and let α=(α1,…,αn)\alpha=(\alpha_1,\dots,\alpha_n) be a multi-index of length nn in the sense of \ref{def:multi-index-length-n-2026a}. The order of α\alpha is the nonnegative integer ∣α∣=α1+⋯+αn.|\alpha|=\alpha_1+\cdots+\alpha_n. The factorial of α\alpha is the natural number α!=α1!⋯αn!,\alpha! = \alpha_1!\cdots \alpha_n!, where each factorial on the right is the one from \reftext{def:factorial-natural-number-2026a}{the factorial definition}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Multi-Index of Length nn

    definitiondef:multi-index-length-n-2026aCombinatoricsMultivariable Calculus
    Let n∈Nn\in\mathbb{N}. A multi-index of length nn is an element α=(α1,…,αn)∈(N∪{0})n.\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}\cup\{0\})^n. That is, a multi-index of length nn is an ordered nn-tuple of nonnegative integers. The zero multi-index of length nn is 0=(0,…,0).0=(0,\dots,0). For each index i∈{1,…,n}i\in\{1,\dots,n\}, the iith standard basis multi-index is ei=(0,…,0,1,0,…,0),e_i=(0,\dots,0,1,0,\dots,0), with 11 in the iith position and 00 elsewhere. If α=(α1,…,αn),β=(β1,…,βn)\alpha=(\alpha_1,\dots,\alpha_n),\qquad \beta=(\beta_1,\dots,\beta_n) are multi-indices of length nn, we write β≤α\beta\le \alpha if and only if βi≤αi\beta_i\le \alpha_i for every i∈{1,…,n}i\in\{1,\dots,n\}. In that case the difference α−β\alpha-\beta is the multi-index of length nn defined componentwise by α−β=(α1−β1,…,αn−βn).\alpha-\beta=(\alpha_1-\beta_1,\dots,\alpha_n-\beta_n). In particular, if αi>0\alpha_i>0, then ei≤αe_i\le \alpha and α−ei=(α1,…,αi−1,αi−1,αi+1,…,αn).\alpha-e_i=(\alpha_1,\dots,\alpha_{i-1},\alpha_i-1,\alpha_{i+1},\dots,\alpha_n).

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

    definitiondef:boundary-smooth-manifold-with-boundary-2026aGeometryTopologyMultivariable Calculus
    Let MM be a smooth manifold with boundary of dimension nn in the sense of \ref{def:smooth-manifold-with-boundary-2026a}, and let p∈Mp\in M. We say that pp is an interior point of MM if there exists a chart (U,φ)(U,\varphi) in the chosen atlas with p∈Up\in U and φ(p)∈int⁡(Hn),\varphi(p)\in \operatorname{int}(H^n), where HnH^n is the half-space from \ref{def:closed-upper-half-space-euclidean-2026a}. We say that pp is a boundary point of MM if there exists a chart (U,φ)(U,\varphi) in the chosen atlas with p∈Up\in U and φ(p)∈∂Hn.\varphi(p)\in \partial H^n. The set of all boundary points of MM is denoted by ∂M\partial M and is called the boundary of MM. The set of all interior points is denoted by int⁡(M).\operatorname{int}(M).

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    Authors ChatGPT-5.4, Aaron · Created

  • Smooth Atlas and Smooth Manifold with Boundary

    definitiondef:smooth-manifold-with-boundary-2026aGeometryTopologyMultivariable Calculus
    Let (X,T)(X,\mathcal{T}) be a \reftext{def:topological-space-2026a}{topological space}, and let n∈Nn\in\mathbb{N}. A smooth atlas of dimension nn on XX, modeled on the closed upper half-space, is a family of charts A=((Uα,φα))α∈A\mathcal{A}=\bigl((U_\alpha,\varphi_\alpha)\bigr)_{\alpha\in A} for some set AA such that the following conditions hold. The chart domains cover XX, that is, X=⋃α∈AUα.X=\bigcup_{\alpha\in A} U_\alpha. For every α,β∈A\alpha,\beta\in A, the charts (Uα,φα)(U_\alpha,\varphi_\alpha) and (Uβ,φβ)(U_\beta,\varphi_\beta) are smoothly compatible in the sense of \ref{def:smooth-compatible-charts-upper-half-space-2026a}. A smooth manifold with boundary of dimension nn is a topological space together with such a smooth atlas, under the additional assumptions that the underlying topological space is \reftext{def:hausdorff-topological-space-2026a}{Hausdorff} and \reftext{def:second-countable-topological-space-2026a}{second countable}. If the underlying topological space is \reftext{def:compact-space-and-subset-2026a}{compact}, then one says that the manifold with boundary is compact.

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    Authors ChatGPT-5.4, Aaron · Created

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

    definitiondef:smooth-compatible-charts-upper-half-space-2026aGeometryTopologyMultivariable Calculus
    Let (X,T)(X,\mathcal{T}) be a topological space, let n∈Nn\in\mathbb{N}, and let (U,φ)(U,\varphi) and (V,ψ)(V,\psi) be charts of dimension nn on XX in the sense of \ref{def:chart-upper-half-space-2026a}. Write φ(U)=Ωφ⊆Hn,ψ(V)=Ωψ⊆Hn.\varphi(U)=\Omega_\varphi\subseteq H^n,\qquad \psi(V)=\Omega_\psi\subseteq H^n. We say that these charts are smoothly compatible if either U∩V=∅U\cap V=\varnothing, or else the following condition holds. For every point a∈φ(U∩V),a\in \varphi(U\cap V), there exist open subsets W,W′⊆RnW,W'\subseteq \mathbb{R}^n with a∈W,(ψ∘φ−1)(a)∈W′,a\in W,\qquad (\psi\circ\varphi^{-1})(a)\in W', and a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} F:W→W′F:W\to W' such that F(x)=(ψ∘φ−1)(x)F(x)=(\psi\circ\varphi^{-1})(x) for every point x∈W∩φ(U∩V).x\in W\cap \varphi(U\cap V). The analogous extension condition is also required for the transition map φ∘ψ−1\varphi\circ\psi^{-1} at every point of ψ(U∩V)\psi(U\cap V).

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  • Chart Modeled on the Closed Upper Half-Space

    definitiondef:chart-upper-half-space-2026aGeometryTopologyMultivariable Calculus
    Let (X,T)(X,\mathcal{T}) be a \reftext{def:topological-space-2026a}{topological space}, let n∈Nn\in\mathbb{N}, and let U⊆XU\subseteq X. A chart of dimension nn on XX, modeled on the closed upper half-space, is a pair (U,φ)(U,\varphi) with the following properties. U∈TU\in\mathcal{T}. If HnH^n denotes the half-space from \reftext{def:closed-upper-half-space-euclidean-2026a}{the definition of the closed upper half-space}, then there exists a subset Ω⊆Hn\Omega\subseteq H^n that is open in HnH^n. φ:U→Ω\varphi:U\to\Omega is a \reftext{def:bijection-sets-2026a}{bijection}. The map φ:U→Ω\varphi:U\to\Omega is \reftext{def:continuous-map-topological-spaces-2026a}{continuous}, where UU carries the \reftext{def:subspace-topology-2026a}{subspace topology} from XX and Ω\Omega carries the subspace topology from HnH^n. The inverse map φ−1:Ω→U\varphi^{-1}:\Omega\to U is continuous for the same topologies. The set UU is called the chart domain, and φ\varphi is called the chart map.

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  • Closed Upper Half-Space in Euclidean Space

    definitiondef:closed-upper-half-space-euclidean-2026aGeometryTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N}. In the \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n, the closed upper half-space is the subset Hn={x=(x1,…,xn)∈Rn:xn≥0}.H^n=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n : x_n\ge 0\}. A subset Ω⊆Hn\Omega\subseteq H^n is said to be open in HnH^n if there exists an \reftext{def:open-subset-euclidean-space-2026a}{open subset} U⊆RnU\subseteq\mathbb{R}^n such that Ω=Hn∩U.\Omega = H^n\cap U. The boundary hyperplane of HnH^n is the subset ∂Hn={x=(x1,…,xn)∈Hn:xn=0}.\partial H^n = \{x=(x_1,\dots,x_n)\in H^n : x_n=0\}. The interior of HnH^n is the subset int⁡(Hn)={x=(x1,…,xn)∈Hn:xn>0}.\operatorname{int}(H^n)=\{x=(x_1,\dots,x_n)\in H^n : x_n>0\}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Open Ball in a Metric Space is Open

    theoremthm:open-ball-metric-space-open-2026aTopology
    Let (X,d)(X,d) be a \reftext{def:metric-space-2026a}{metric space}, let x∈Xx\in X, and let r∈Rr\in\mathbb{R} satisfy r>0r>0. Then the \reftext{def:open-ball-metric-space-2026a}{open ball} Bd(x,r)B_d(x,r) is \reftext{def:open-subset-metric-space-2026a}{open} in the metric space (X,d)(X,d).

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    Authors ChatGPT-5.4, Aaron · Created

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

    theoremthm:heine-borel-rn-2026aAnalysisTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, and let A⊆RnA\subseteq\mathbb{R}^n. Then the following are equivalent. AA is \reftext{def:compact-space-and-subset-2026a}{compact in Rn\mathbb{R}^n}, where Rn\mathbb{R}^n is regarded as a topological space through the topology determined by the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}. AA is \reftext{def:closed-subset-topological-space-2026a}{closed} in Rn\mathbb{R}^n and \reftext{def:bounded-subset-metric-space-2026a}{bounded} as a subset of the metric space (Rn,dE)(\mathbb{R}^n,d_E).

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    Authors ChatGPT-5.4, Aaron · Created

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

    theoremthm:compact-subset-rn-closed-2026aAnalysisTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, and let A⊆RnA\subseteq\mathbb{R}^n. Assume that AA is \reftext{def:compact-space-and-subset-2026a}{compact in Rn\mathbb{R}^n}, where Rn\mathbb{R}^n is regarded as a topological space through the topology determined by the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}. Then AA is \reftext{def:closed-subset-topological-space-2026a}{closed} in Rn\mathbb{R}^n.

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    Authors ChatGPT-5.4, Aaron · Created

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

    theoremthm:compact-subset-rn-bounded-2026aAnalysisTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, and let A⊆RnA\subseteq\mathbb{R}^n. Assume that AA is \reftext{def:compact-space-and-subset-2026a}{compact in Rn\mathbb{R}^n}, where Rn\mathbb{R}^n is regarded as a topological space through the topology determined by the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}. Then AA is \reftext{def:bounded-subset-metric-space-2026a}{bounded} as a subset of the metric space (Rn,dE)(\mathbb{R}^n,d_E).

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    Authors ChatGPT-5.4, Aaron · Created

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

    theoremthm:closed-box-compact-rn-2026aAnalysisTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N}. For each index i∈{1,…,n}i\in\{1,\dots,n\}, let ai,bi∈Ra_i,b_i\in\mathbb{R} satisfy ai≤bia_i\le b_i, and let B⊆RnB\subseteq\mathbb{R}^n be the \reftext{def:closed-box-rn-2026a}{closed box} determined by these endpoints. Then BB is \reftext{def:compact-space-and-subset-2026a}{compact in Rn\mathbb{R}^n}, where Rn\mathbb{R}^n is regarded as a topological space through the topology determined by the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}.

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    Authors ChatGPT-5.4, Aaron · Created

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

    theoremthm:euclidean-open-box-criterion-rn-2026aTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N} and let U⊆RnU\subseteq \mathbb{R}^n. Then UU is \reftext{def:open-subset-euclidean-space-2026a}{open in the Euclidean sense} if and only if for every point x=(x1,…,xn)∈Ux=(x_1,\dots,x_n)\in U there exists a real number δ>0\delta>0 such that every point y=(y1,…,yn)∈Rny=(y_1,\dots,y_n)\in\mathbb{R}^n satisfying xi−δ<yi<xi+δx_i-\delta<y_i<x_i+\delta for every i∈{1,…,n}i\in\{1,\dots,n\} belongs to UU.

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    Authors ChatGPT-5.4, Aaron · Created

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

    theoremthm:closed-interval-compact-real-2026aAnalysisTopology
    Let a,b∈Ra,b\in\mathbb{R} satisfy a≤ba\le b. Then the interval [a,b][a,b] from \reftext{def:interval-real-line-c54-2026c}{the interval definition} is \reftext{def:compact-space-and-subset-2026a}{compact in R\mathbb{R}}, where R\mathbb{R} is regarded as a topological space through the topology determined by the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Closed Box in Rn\mathbb{R}^n

    definitiondef:closed-box-rn-2026aTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N}. For each index i∈{1,…,n}i\in\{1,\dots,n\}, let ai,bi∈Ra_i,b_i\in\mathbb{R} satisfy ai≤bia_i\le b_i. The subset B={x=(x1,…,xn)∈Rn:ai≤xi≤bi for every i∈{1,…,n}}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 Rn\mathbb{R}^n determined by the endpoints a1,b1,…,an,bna_1,b_1,\dots,a_n,b_n.

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    Authors ChatGPT-5.4, Aaron · Created

  • Euclidean Openness Agrees with Metric Openness on Rn\mathbb{R}^n

    theoremthm:euclidean-open-iff-metric-open-rn-2026aTopologyMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n, and let dEd_E be the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} on Rn\mathbb{R}^n. Then UU is \reftext{def:open-subset-euclidean-space-2026a}{open in the Euclidean sense} if and only if UU is \reftext{def:open-subset-metric-space-2026a}{open in the metric space (Rn,dE)(\mathbb{R}^n,d_E)}.

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