Theorems

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

Showing 21-40 of 124

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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
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
Sequential Characterization of Closed Subsets of a Metric Space
lemmalem:sequentially-closed-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 $A\subseteq X$ , and let $\mathbb{N}$ denote the natural numbers. Then $A$ is closed in the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T19:47:44.665471Z
A Subsequence of a Convergent Sequence Has the Same Limit
lemmalem:subsequence-convergent-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ , and let $x\in X$ be such that $(x_m)_{m\in\mathbb{N}}$ converges to $x$ in $(X,d)$ . Let $(n_k)_{k\in\mathbb{N}}$ be a strictly increasing sequence in $\mathbb{N}$ , so that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T06:34:17.735432Z
The Product Topology is a Topology
lemmalem:product-topology-is-topology-2026aTopology
Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces, let $X\times Y$ be their Cartesian product, and let $\mathcal{T}_{X\times Y}$ be the product topology on $X\times Y$ . Then $(X\times Y,\mathcal{T}_{X\times Y})$ is a topological space.
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T05:08:54.507812Z
Continuous Image of a Compact Space is Compact
theoremthm:continuous-image-compact-is-compact-2026bTopology
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 $Y$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T05:08:22.318247Z
Compactness in a Subspace Agrees with Compactness in the Ambient Space
lemmalem:compact-in-subspace-iff-ambient-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, let $B\subseteq X$ , and let $\mathcal{T}_B=\{B\cap U : U\in\mathcal{T}\}$ be the subspace topology on $B$ , so that $(B,\mathcal{T}_B)$ is a topological space by The Subspace Topology is a Topology. Let $A\subseteq B$ . Then the fol…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T05:08:01.688951Z
Heine-Borel Theorem in $\mathbb{R}^n$
theoremthm:heine-borel-rn-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 2026-08-16T04:24:14.074763Z
Closed Subset of a Compact Space is Compact
theoremthm:closed-subset-compact-is-compact-2026bTopology
Let $(X,\mathcal{T})$ be a topological space that is compact, and let $A\subseteq X$ be closed in $X$ . Then $A$ is compact in $X$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T04:23:35.796658Z
Compact Subset Criterion via Open Covers in the Ambient Space
theoremthm:compact-subset-open-cover-criterion-2026bTopology
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 exists a finite subset $J\subseteq I$ such that the subfamily…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T04:21:16.912831Z
The Subspace Topology is a Topology
lemmalem:subspace-topology-is-topology-2026aTopology
Let $(X,\mathcal{T})$ be a topological space, let $A\subseteq X$ , and let $\mathcal{T}_A$ be the subspace topology on $A$ . Then $(A,\mathcal{T}_A)$ is a topological space.
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T04:20:46.091870Z
Compact Topological Space and Compact Subset
definitiondef:compact-space-and-subset-2026bTopology
Let $(X,\mathcal{T})$ be a topological space. We say that $X$ is compact if for every set $I$ and every family of subsets of $X$ $(U_i)_{i\in I}$ such that $U_i\in\mathcal{T}$ for every $i\in I$ and $X\subseteq\bigcup_{i\in I}U_i,$ there exists a finite subset $J\subseteq I$ …
+2 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T03:49:55.519352Z
Compactness and Sequential Compactness Agree for Subsets of a Metric Space
corollarycor:compact-iff-sequentially-compact-metric-2026bAnalysis Topology
Let $(X,d)$ be a metric space, let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology, and let $K\subseteq X$ . Then $K$ is compact in $(X,\mathcal{T}_d)$ if and only if $K$ is…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T03:31:34.164599Z
A Sequentially Compact Subset of a Metric Space is Compact
theoremthm:sequentially-compact-implies-compact-metric-2026bAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be sequentially compact in $(X,d)$ . Then $K$ is compact in $(X,\mathcal{T}_d)$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T03:31:04.129798Z
A Sequentially Compact Subset of a Metric Space is Totally Bounded
theoremthm:sequentially-compact-implies-totally-bounded-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $K\subseteq X$ be sequentially compact in $(X,d)$ . Then for every real number $\varepsilon>0$ there exists a finite subset $F\subseteq K$ such that $K\subseteq\bigcup_{a\in F}B_d(a,\varepsilon),$ where $B_d(a,\varepsilon)$ is the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T03:29:47.160946Z
Lebesgue Number Lemma for a Sequentially Compact Subset of a Metric Space
lemmalem:lebesgue-number-sequentially-compact-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be sequentially compact in $(X,d)$ , let $I$ be a set, and let $(U_i)_{i\in I}$ b…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T03:29:11.051742Z
A Compact Subset of a Metric Space is Totally Bounded
theoremthm:compact-implies-totally-bounded-metric-2026bAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_d)$ . Then $K$ is totally bounded in $(X,d)$ .
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T03:01:26.249818Z
Totally Bounded Subset of a Metric Space
definitiondef:totally-bounded-subset-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $K\subseteq X$ . We say that $K$ is totally bounded in $(X,d)$ if for every real number $\varepsilon>0$ there exists a finite subset $F\subseteq X$ such that $K\subseteq\bigcup_{a\in F}B_d(a,\varepsilon),$ where $B_d(a,\varepsilon)$ deno…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T03:01:19.229360Z
A Compact Subset of a Metric Space is Sequentially Compact
corollarycor:compact-implies-sequentially-compact-metric-2026bAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_d)$ . Then $K$ is sequentially compact in $(X,d)$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:45:12.559902Z

Showing 21-40 of 124

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Uniqueness of Limits in a Metric Space

A Closed Interval is Sequentially Compact in the Real Line

A Totally Bounded Subset of a Nonempty Metric Space is Bounded

Sequential Characterization of Closed Subsets of a Metric Space

A Subsequence of a Convergent Sequence Has the Same Limit

The Product Topology is a Topology

Continuous Image of a Compact Space is Compact

Compactness in a Subspace Agrees with Compactness in the Ambient Space

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

Closed Subset of a Compact Space is Compact

Compact Subset Criterion via Open Covers in the Ambient Space

The Subspace Topology is a Topology

Compact Topological Space and Compact Subset

Compactness and Sequential Compactness Agree for Subsets of a Metric Space

A Sequentially Compact Subset of a Metric Space is Compact

A Sequentially Compact Subset of a Metric Space is Totally Bounded

Lebesgue Number Lemma for a Sequentially Compact Subset of a Metric Space

A Compact Subset of a Metric Space is Totally Bounded

Totally Bounded Subset of a Metric Space

A Compact Subset of a Metric Space is Sequentially Compact

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