Theorems

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

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Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$
theoremthm:euclidean-open-iff-metric-open-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ , and let $d_E$ be the Euclidean distance on $\mathbb{R}^n$ . Then $U$ is open in the Euclidean sense if and only if $U$ is open in the metric space $(\mathbb{R}^n,d_E)$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:24.013485Z
Euclidean Distance is a Metric on $\mathbb{R}^n$
theoremthm:euclidean-distance-is-metric-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ . Then the Euclidean distance $d_E$ is a metric on $\mathbb{R}^n$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.974349Z
Euclidean Distance on $\mathbb{R}^n$
definitiondef:euclidean-distance-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ . Consider the set $\mathbb{R}^n$ appearing in the definition of open subsets of Euclidean space. For points $x=(x_1,\dots,x_n),\qquad y=(y_1,\dots,y_n)$ in $\mathbb{R}^n$ , define $d_E(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2},$ where $\sqrt{\ \cdot\ }$ den…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.927597Z
Metric Open Sets Form a Topology
theoremthm:metric-open-sets-form-topology-2026aTopology
Let $X$ be a set, let $d$ be a metric on $X$ , and let $\mathcal{T}_d$ be the collection of all subsets $U\subseteq X$ that are open in the metric space $(X,d)$ . Then $(X,\mathcal{T}_d)$ is a topological space.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.849523Z
Bounded Subset of a Metric Space
definitiondef:bounded-subset-metric-space-2026aTopology
Let $(X,d)$ be a metric space, and let $A\subseteq X$ . We say that $A$ is bounded in $(X,d)$ if there exist a point $x\in X$ and a real number $R>0$ such that $d(x,y)\le R$ for every $y\in A$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.804928Z
Open Subset of a Metric Space
definitiondef:open-subset-metric-space-2026aTopology
Let $(X,d)$ be a metric space, and let $U\subseteq X$ . We say that $U$ is open in the metric space $(X,d)$ if for every point $x\in U$ there exists a real number $r>0$ such that $B_d(x,r)\subseteq U,$ where $B_d(x,r)$ is the open ball with center $x$ and radius $r$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.757424Z
Open Ball in a Metric Space
definitiondef:open-ball-metric-space-2026aTopology
Let $(X,d)$ be a metric space, let $x\in X$ , and let $r\in\mathbb{R}$ satisfy $r>0$ . The open ball in $X$ with center $x$ and radius $r$ is the subset $B_d(x,r)=\{y\in X:d(x,y)<r\}.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.693530Z
Metric Space
definitiondef:metric-space-2026aTopology
Let $X$ be a set, and let $d:X\times X\to\mathbb{R}$ . We say that $d$ is a metric on $X$ if for every $x,y,z\in X$ the following conditions hold. 1. $0\le d(x,y)$ . 2. $d(x,y)=0$ if and only if $x=y$ . 3. $d(x,y)=d(y,x)$ . 4. $d(x,z)\le d(x,y)+d(y,z)$ . If $d$ is a metric on $X$ , t…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.634275Z
Existence and Uniqueness of the Nonnegative Square Root
theoremthm:nonnegative-real-has-unique-square-root-2026aAnalysis
Let $a\in\mathbb{R}$ satisfy $0\le a$ . Then there exists a unique real number $r\in\mathbb{R}$ such that $0\le r$ and $r^2=a$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.491180Z
Compact Subset Criterion via Open Covers in the Ambient Space
theoremthm:compact-subset-open-cover-criterion-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . Then the following are equivalent. 1. The subset $A$ is compact in $X$ . 2. For every open cover $(U_i)_{i\in I}$ of $A$ in $X$ , there exist a natural number $n\in\mathbb{N}$ and elements $i_1,\dots,i_n\in I$ s…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T18:19:05.099659Z
Open Cover and Subcover of a Subset of a Topological Space
definitiondef:open-cover-subcover-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, let $A\subseteq X$ , let $I$ be a set, and let $(U_i)_{i\in I}$ be a family of subsets of $X$ . We say that $(U_i)_{i\in I}$ is an open cover of $A$ in $X$ if the following two conditions hold. 1. For every $i\in I$ , one has…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T18:19:05.015588Z
Family and Subfamily of Subsets of a Set
definitiondef:family-subfamily-subsets-set-2026aTopology Set Theory
Let $X$ be a set, let $A$ be a set, and suppose that for each element $a\in A$ a subset $U_a\subseteq X$ is specified. The collection $(U_a)_{a\in A}$ is called a family of subsets of $X$ indexed by $A$ . If $B\subseteq A$ , then the collection $(U_b)_{b\in B}$ is called the subfa…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T18:04:07.322498Z
Complement of a Subset Relative to a Set
definitiondef:complement-subset-relative-set-2026aTopology Set Theory
Let $X$ be a set, and let $A\subseteq X$ . The complement of $A$ relative to $X$ is the subset of $X$ defined by $X\setminus A = \{x\in X : x\notin A\}.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T18:04:07.246219Z
Product of Two Compact Spaces is Compact
theoremthm:product-two-compact-spaces-2026aTopology
Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces. Suppose that both $X$ and $Y$ are compact. Then $X\times Y$ is compact when equipped with the product topology.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:07.155108Z
Continuous Image of a Compact Space is Compact
theoremthm:continuous-image-compact-is-compact-2026aTopology
Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces, and let $f:X\to Y$ be a continuous map. If $X$ is compact, then the image $f(X)=\{f(x):x\in X\}$ is compact in $Y$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:07.116531Z
Closed Subset of a Compact Space is Compact
theoremthm:closed-subset-compact-is-compact-2026aTopology
Let $(X,\mathcal{T})$ be a topological space that is compact, and let $A\subseteq X$ be closed. Then $A$ is compact in $X$ , in the sense of the definition of compact subset.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:07.076261Z
Product Topology
definitiondef:product-topology-2026aTopology
Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces. The product topology on the Cartesian product $X\times Y$ is the collection of all subsets $W\subseteq X\times Y$ with the following property: for every point $(x,y)\in W$ , there exist sets…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:06.984253Z
Subspace Topology
definitiondef:subspace-topology-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . The subspace topology on $A$ is the collection $\mathcal{T}_A = \{A\cap U : U\in\mathcal{T}\}.$ The sets in $\mathcal{T}_A$ are called the open sets of $A$ in the subspace topology.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:06.919371Z
Continuous Map Between Topological Spaces
definitiondef:continuous-map-topological-spaces-2026aTopology
Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces, and let $f:X\to Y$ be a function. We say that $f$ is continuous if for every open set $V\in\mathcal{T}_Y$ , the inverse image $f^{-1}(V)=\{x\in X : f(x)\in V\}$ belongs to $\mathcal{T}_X$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:06.872808Z
Closed Subset of a Topological Space
definitiondef:closed-subset-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . We say that $A$ is closed in $X$ if its complement relative to $X$ is open, that is, if $X\setminus A\in\mathcal{T}$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:06.817233Z

Showing 1381-1400 of 1477

Back Next

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

Euclidean Distance is a Metric on Rn\mathbb{R}^nRn

Euclidean Distance on Rn\mathbb{R}^nRn

Metric Open Sets Form a Topology

Bounded Subset of a Metric Space

Open Subset of a Metric Space

Open Ball in a Metric Space

Metric Space

Existence and Uniqueness of the Nonnegative Square Root

Compact Subset Criterion via Open Covers in the Ambient Space

Open Cover and Subcover of a Subset of a Topological Space

Family and Subfamily of Subsets of a Set

Complement of a Subset Relative to a Set

Product of Two Compact Spaces is Compact

Continuous Image of a Compact Space is Compact

Closed Subset of a Compact Space is Compact

Product Topology

Subspace Topology

Continuous Map Between Topological Spaces

Closed Subset of a Topological Space

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

Euclidean Distance is a Metric on $\mathbb{R}^n$

Euclidean Distance on $\mathbb{R}^n$