Theorems

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

Showing 1-20 of 302

Continuity of a Map Between Metric Spaces via Preimages of Open Sets
theoremthm:continuity-preimage-open-metric-2026aAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Let $\mathcal{T}_{d_X}$ be the collection of all subsets of $X$ that are open in $(X,d_X)$ and let $\mathcal{T}_{d_Y}$ be the collection of all subsets of $Y$ open in $(Y,d_Y)$ ; both are topologies by Metric Open Sets Form a Topology.…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-17T03:40:28.989324Z
Vanishing of the Distance to a Set Characterizes the Closure
lemmalem:distance-to-set-zero-closure-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of all subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $A\subseteq X$ be nonempty and let $x\in X$ , and write $\operatorname{dist}_d(x,A)$ for the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-17T03:40:25.440614Z
The Distance to a Set is Nonexpansive
lemmalem:distance-to-set-lipschitz-2026aAnalysis Topology
Let $(X,d)$ be a metric space and let $A\subseteq X$ be nonempty, and write $\operatorname{dist}_d(z,A)$ for the distance from a point $z\in X$ to $A$ in $(X,d)$ . Let $\mathbb{R}$ denote the real numbers, with the order, addition and additive inverses of their ordered field struc…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-17T03:40:22.746075Z
Distance from a Point to a Nonempty Subset of a Metric Space
definitiondef:distance-point-to-set-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $A\subseteq X$ be nonempty, and let $x\in X$ . Let $S_{x,A}=\{t\in\mathbb{R}:\ t=d(x,a) \text{ for some } a\in A\},$ where $\mathbb{R}$ denotes the real numbers. Then $S_{x,A}$ is nonempty because $A$ is, and $0$ is a lower bound for…
+0 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-17T03:39:06.889353Z
Uniformly Continuous Map Between Metric Spaces
definitiondef:uniformly-continuous-metric-2026aAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $A\subseteq X$ , and let $f:A\to Y$ . Let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure, and for $a,b\in\mathbb{R}$ write $a<b$ to mean that $a\le b$ and $a\ne b$ . We say that $f$ is…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-17T03:39:02.691615Z
The Closure of a Bounded Subset of $\mathbb{R}^n$ is Compact
corollarycor:closure-bounded-rn-compact-2026aAnalysis Topology Multivariable Calculus
Let $n$ be a natural number, let $d_E$ be the Euclidean distance on Euclidean space $\mathbb{R}^n$ , which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$ , and let $\mathcal{T}_{d_E}$ be the collection of subsets of $\mathbb{R}^n$ that are…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T23:07:58.394884Z
The Closure of a Bounded Subset of a Metric Space is Bounded
lemmalem:closure-bounded-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of all subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $A\subseteq X$ be bounded in $(X,d)$ . Then the closure $\operatorname{cl}_X(A)$ of $A$ in…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T23:07:53.064075Z
Sequential Characterization of the Closure in a Metric Space
lemmalem:closure-sequential-characterization-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of all subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $A\subseteq X$ , let $x\in X$ , and let $\mathbb{N}$ denote the natural numbers. Then $x$ be…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T23:07:49.552726Z
Characterization of the Closure in a Metric Space by Open Balls
theoremthm:closure-metric-characterization-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of all subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $A\subseteq X$ and let $x\in X$ . Then the following are equivalent. 1. The point $x$ belo…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T23:07:44.696277Z
Closed Interval $[a,b]$ is Compact in $\mathbb{R}$
theoremthm:closed-interval-compact-real-2026bAnalysis Topology
Let $a,b$ be real numbers with $a\le b$ in the order of the ordered field $\mathbb{R}$ , and let $[a,b]$ be the closed interval determined by $a$ and $b$ . Let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, whose metric coincides with the Euclidean distance on $\mathbb{R}^{1}$ id…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T20:26:14.254294Z
Compact Subset of $\mathbb{R}^n$ is Bounded
theoremthm:compact-subset-rn-bounded-2026bAnalysis Topology Multivariable Calculus
Let $n$ be a natural number, let $d_E$ be the Euclidean distance on Euclidean space $\mathbb{R}^n$ , which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$ , and let $\mathcal{T}_{d_E}$ be the collection of subsets of $\mathbb{R}^n$ that are…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T20:24:09.284691Z
Compact Subset of $\mathbb{R}^n$ is Closed
theoremthm:compact-subset-rn-closed-2026bAnalysis Topology Multivariable Calculus
Let $n$ be a natural number, let $d_E$ be the Euclidean distance on Euclidean space $\mathbb{R}^n$ , which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$ , and let $\mathcal{T}_{d_E}$ be the collection of subsets of $\mathbb{R}^n$ that are…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T20:23:53.103322Z
Uniqueness of Limits in a Metric Space
lemmalem:limit-unique-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ , and let $x,y\in X$ . If $(x_m)_{m\in\mathbb{N}}$ converges to $x$ in $(X,d)$ and also converges to $y$ in $(X,d)$ , then $x=y$ .
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T20:23:20.416817Z
Bolzano-Weierstrass Theorem in Euclidean Space
theoremthm:bolzano-weierstrass-rn-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number and let $\mathbb{R}^n$ be Euclidean space equipped with the Euclidean distance $d_E$ , which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$ . Let $K\subseteq\mathbb{R}^n$ be bounded in $(\mathbb{R}^n,d_E)$ , and let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T20:08:21.029625Z
Convergence in Euclidean Space is Coordinatewise Convergence
lemmalem:convergence-coordinatewise-rn-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number, let $[n]$ be the initial segment of $\mathbb{N}$ determined by $n$ , and let $\mathbb{R}^n$ be Euclidean space equipped with the Euclidean distance $d_E$ , which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$ . Let…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T20:06:49.174483Z
Coordinate Bounds Control the Euclidean Norm
lemmalem:euclidean-norm-coordinate-bound-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number, let $[n]$ be the initial segment of $\mathbb{N}$ determined by $n$ , and let $x=(x_1,\dots,x_n)$ be a point of Euclidean space $\mathbb{R}^n$ . Write $\lVert\,\cdot\,\rVert$ for the Euclidean norm, $|\cdot|$ for the absolute value on the real numbers, w…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T20:05:45.036292Z
The Archimedean Property of the Real Numbers
theoremthm:archimedean-property-real-2026aAnalysis
Let $\mathbb{R}$ denote the real numbers, with the addition, multiplication, additive identity $0$ and multiplicative inverses $a^{-1}$ of the underlying field and with the order $\le$ of its ordered field structure; for $s,t\in\mathbb{R}$ write $s<t$ to mean $s\le t$ and…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T20:04:04.426897Z
Properties of the Canonical Map from the Natural Numbers to an Ordered Field
lemmalem:natural-number-image-properties-2026aAnalysis Algebra
Let $F$ be an ordered field, with the addition, multiplication, additive identity $0$ , multiplicative identity $1$ and multiplicative inverses $a^{-1}$ of the underlying field, and with its order $\le$ ; for $a,b\in F$ write $a<b$ to mean $a\le b$ and $a\ne b$ . Let $\mathbb{N}$ be…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T20:02:46.097843Z
A Closed Interval is Sequentially Compact in the Real Line
theoremthm:closed-interval-sequentially-compact-real-2026aAnalysis Topology
Let $\mathbb{R}$ denote the real numbers, with the order $\le$ of its ordered field structure, and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $a,b\in\mathbb{R}$ satisfy $a\le b$ , and let $[a,b]$ be the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T19:52:12.376949Z
A Totally Bounded Subset of a Nonempty Metric Space is Bounded
lemmalem:totally-bounded-implies-bounded-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space whose underlying set $X$ is nonempty, and let $K\subseteq X$ be totally bounded in $(X,d)$ . Then $K$ is bounded in $(X,d)$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T19:49:39.539220Z

Showing 1-20 of 302

Continuity of a Map Between Metric Spaces via Preimages of Open Sets

Vanishing of the Distance to a Set Characterizes the Closure

The Distance to a Set is Nonexpansive

Distance from a Point to a Nonempty Subset of a Metric Space

Uniformly Continuous Map Between Metric Spaces

The Closure of a Bounded Subset of Rn\mathbb{R}^nRn is Compact

The Closure of a Bounded Subset of a Metric Space is Bounded

Sequential Characterization of the Closure in a Metric Space

Characterization of the Closure in a Metric Space by Open Balls

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

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

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

Uniqueness of Limits in a Metric Space

Bolzano-Weierstrass Theorem in Euclidean Space

Convergence in Euclidean Space is Coordinatewise Convergence

Coordinate Bounds Control the Euclidean Norm

The Archimedean Property of the Real Numbers

Properties of the Canonical Map from the Natural Numbers to an Ordered Field

A Closed Interval is Sequentially Compact in the Real Line

A Totally Bounded Subset of a Nonempty Metric Space is Bounded

The Closure of a Bounded Subset of $\mathbb{R}^n$ is Compact

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

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

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