Theorems

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

Showing 1-20 of 124

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
Decomposition of a Topological Space by the Boundary of a Subset
lemmalem:boundary-decomposition-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . Write $X\setminus S$ for the complement relative to $X$ of a subset $S\subseteq X$ , let $\operatorname{int}_X$ and $\operatorname{cl}_X$ denote the interior and the closure in $X$ , and let $\partial_X$ denote t…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T23:07:29.103458Z
Boundary of a Subset of a Topological Space
definitiondef:boundary-subset-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . The boundary of $A$ in $X$ is the subset $\partial_X A=\operatorname{cl}_X(A)\setminus\operatorname{int}_X(A)$ of $X$ , where $\operatorname{cl}_X(A)$ is the closure of $A$ in $X$ and…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T23:07:20.859168Z
The Closure is the Smallest Closed Superset
theoremthm:closure-smallest-closed-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . Let $\operatorname{cl}_X(A)$ denote the closure of $A$ in $X$ , and call a subset of $X$ closed when it is closed in the topological space $(X,\mathcal{T})$ . Then the following hold. 1.…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T23:07:00.631299Z
The Interior is the Largest Open Subset
theoremthm:interior-largest-open-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . Let $\operatorname{int}_X(A)$ denote the interior of $A$ in $X$ . Then the following hold. 1. $\operatorname{int}_X(A)\subseteq A$ . 2. $\operatorname{int}_X(A)\in\mathcal{T}$ . 3. If $U\in\mathcal{T}$ and…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T23:06:56.244865Z
Duality Between Interior and Closure Under Complementation
lemmalem:interior-closure-complement-duality-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . Write $X\setminus S$ for the complement relative to $X$ of a subset $S\subseteq X$ , and let $\operatorname{int}_X$ and $\operatorname{cl}_X$ denote the interior and the closure in $X$ . Then the following hold.…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T23:06:52.006793Z
Closure of a Subset of a Topological Space
definitiondef:closure-subset-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . The closure of $A$ in $X$ is the subset $\operatorname{cl}_X(A)=\{x\in X:\ U\cap A\neq\varnothing \text{ for every } U\in\mathcal{T} \text{ with } x\in U\}$ of $X$ . Its elements are called the adherent poin…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T23:06:28.238336Z
Interior of a Subset of a Topological Space
definitiondef:interior-subset-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$ . The interior of $A$ in $X$ is the subset $\operatorname{int}_X(A)=\{x\in X:\ \text{there is } U\in\mathcal{T} \text{ with } x\in U \text{ and } U\subseteq A\}$ of $X$ . Its elements are called the interior p…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T23:06:25.789724Z
Complements, Unions and Intersections of Closed Sets in a Topological Space
lemmalem:closed-sets-topology-2026aTopology
Let $(X,\mathcal{T})$ be a topological space. For a subset $S\subseteq X$ write $X\setminus S$ for the complement of $S$ relative to $X$ , so that $S$ is closed in $X$ exactly when $X\setminus S\in\mathcal{T}$ . Let $\mathbb{N}$ denote the natural numbers. Then the following hold.…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T23:06:22.249966Z
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

Showing 1-20 of 124

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

Decomposition of a Topological Space by the Boundary of a Subset

Boundary of a Subset of a Topological Space

The Closure is the Smallest Closed Superset

The Interior is the Largest Open Subset

Duality Between Interior and Closure Under Complementation

Closure of a Subset of a Topological Space

Interior of a Subset of a Topological Space

Complements, Unions and Intersections of Closed Sets in a Topological Space

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

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