Theorems

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

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Euclidean Distance is a Metric on $\mathbb{R}^n$
theoremthm:euclidean-distance-is-metric-rn-2026aTopologyMultivariable Calculus
Let $n\in\mathbb{N}$ . Then the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} $d_E$ is a \reftext{def:metric-space-2026a}{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-2026aTopologyMultivariable Calculus
Let $n\in\mathbb{N}$ . Consider the set $\mathbb{R}^n$ appearing in \reftext{def:open-subset-euclidean-space-2026a}{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 $\sqrt{\ \cdot\ }$ denotes the nonnegative square root from \ref{thm:nonnegative-real-has-unique-square-root-2026a}. This is a function on the \reftext{def:cartesian-product-sets-2026a}{Cartesian product} $\mathbb{R}^n\times\mathbb{R}^n$ with values in $\mathbb{R}$ , and it is called the Euclidean distance on $\mathbb{R}^n$ .
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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 \reftext{def:metric-space-2026a}{metric} on $X$ , and let $\mathcal{T}_d$ be the collection of all subsets $U\subseteq X$ that are \reftext{def:open-subset-metric-space-2026a}{open in the metric space $(X,d)$ }. Then $(X,\mathcal{T}_d)$ is a \reftext{def:topological-space-2026a}{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 \reftext{def:metric-space-2026a}{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 \reftext{def:metric-space-2026a}{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 \reftext{def:open-ball-metric-space-2026a}{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 \reftext{def:metric-space-2026a}{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. $0\le d(x,y)$ . $d(x,y)=0$ if and only if $x=y$ . $d(x,y)=d(y,x)$ . $d(x,z)\le d(x,y)+d(y,z)$ . If $d$ is a metric on $X$ , then the pair $(X,d)$ is called a metric space.
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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 \reftext{def:topological-space-2026a}{topological space}, and let $A\subseteq X$ . Then the following are equivalent. The subset $A$ is \reftext{def:compact-space-and-subset-2026a}{compact in $X$ }. For every \reftext{def:open-cover-subcover-topological-space-2026a}{open cover} $(U_i)_{i\in I}$ of $A$ in $X$ , there exist a \reftext{def:natural-numbers-2026a}{natural number} $n\in\mathbb{N}$ and elements $i_1,\dots,i_n\in I$ such that $(U_{i_1},\dots,U_{i_n})$ is a finite subcover of $A$ , that is, $A\subseteq U_{i_1}\cup\cdots\cup U_{i_n}.$
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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 \reftext{def:topological-space-2026a}{topological space}, let $A\subseteq X$ , let $I$ be a set, and let $(U_i)_{i\in I}$ be a \reftext{def:family-subfamily-subsets-set-2026a}{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. For every $i\in I$ , one has $U_i\in\mathcal{T}$ . One has $A\subseteq \bigcup_{i\in I} U_i.$ If $(U_i)_{i\in I}$ is an open cover of $A$ in $X$ and if $J\subseteq I$ , then the \reftext{def:family-subfamily-subsets-set-2026a}{subfamily} $(U_j)_{j\in J}$ is called a subcover of $A$ if $A\subseteq \bigcup_{j\in J} U_j.$
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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-2026aTopologySet 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 subfamily of $(U_a)_{a\in A}$ indexed by $B$ .
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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-2026aTopologySet 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 \reftext{def:topological-space-2026a}{topological spaces}. Suppose that both $X$ and $Y$ are \reftext{def:compact-space-and-subset-2026a}{compact}. Then $X\times Y$ is compact when equipped with the \reftext{def:product-topology-2026a}{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 \reftext{def:topological-space-2026a}{topological spaces}, and let $f:X\to Y$ be a \reftext{def:continuous-map-topological-spaces-2026a}{continuous map}. If $X$ is \reftext{def:compact-space-and-subset-2026a}{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 \reftext{def:topological-space-2026a}{topological space} that is \reftext{def:compact-space-and-subset-2026a}{compact}, and let $A\subseteq X$ be \reftext{def:closed-subset-topological-space-2026a}{closed}. Then $A$ is compact in $X$ , in the sense of the \reftext{def:compact-space-and-subset-2026a}{definition of compact subset}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:07.076261Z
Compact Topological Space and Compact Subset
definitiondef:compact-space-and-subset-2026aTopology
Let $(X,\mathcal{T})$ be a \reftext{def:topological-space-2026a}{topological space}. We say that $X$ is compact if for every set $A$ and every \reftext{def:family-subfamily-subsets-set-2026a}{family of subsets of $X$ } $(U_a)_{a\in A}$ such that $U_a\in\mathcal{T}$ for every $a\in A$ and $X\subseteq \bigcup_{a\in A} U_a,$ there exist a \reftext{def:natural-numbers-2026a}{natural number} $n\in\mathbb{N}$ and elements $a_1,\dots,a_n\in A$ such that $X\subseteq U_{a_1}\cup\cdots\cup U_{a_n}.$ If $A\subseteq X$ , we say that $A$ is compact in $X$ if $A$ is compact as a topological space equipped with the \reftext{def:subspace-topology-2026a}{subspace topology}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:07.036464Z
Product Topology
definitiondef:product-topology-2026aTopology
Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be \reftext{def:topological-space-2026a}{topological spaces}. The product topology on the \reftext{def:cartesian-product-sets-2026a}{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 $U\in\mathcal{T}_X$ and $V\in\mathcal{T}_Y$ such that $(x,y)\in U\times V\subseteq W.$
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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 \reftext{def:topological-space-2026a}{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 \reftext{def:topological-space-2026a}{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 \reftext{def:topological-space-2026a}{topological space}, and let $A\subseteq X$ . We say that $A$ is closed in $X$ if its \reftext{def:complement-subset-relative-set-2026a}{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 121-140 of 321

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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

Compact Topological Space and Compact Subset

Product Topology

Subspace Topology

Continuous Map Between Topological Spaces

Closed Subset of a Topological Space

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

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