query: Compactness of a set : $A=\left\lbrace 2019,1010,\ldots,(n+2018)/n,\ldots \right\rbrace$ and $B=A \cup \left\lbrace 1 \right\rbrace$
positive_passages.docid: 28391
positive_passages.text: Prove the compactness of a set B
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 116527
negative_passages.text: Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
negative_passages.docid: 83996
negative_passages.text: For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
negative_passages.docid: 26215
negative_passages.text: Proof Verification: Let $D=\{ \frac{1}{n} \mid n\in\mathbb{N} \} \cup\{0\} \subset \mathbb{E}^1$. Prove that $D$ is compact.
negative_passages.docid: 13167
negative_passages.text: Prove $A = (A \setminus B) \cup (A \cap B)$
negative_passages.docid: 155999
negative_passages.text: How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$
negative_passages.docid: 18596
negative_passages.text: A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition
negative_passages.docid: 88904
negative_passages.text: Help with a proof. Countable sets.
negative_passages.docid: 154978
negative_passages.text: Some equivalent formulations of compactness of a metric space
negative_passages.docid: 3708
negative_passages.text: How can I prove that compact subset iff sequentially compact in second countable topological spaces?
negative_passages.docid: 12472
negative_passages.text: Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable
negative_passages.docid: 23442
negative_passages.text: Cardinality of the union of infinite and countable sets
negative_passages.docid: 65358
negative_passages.text: Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
negative_passages.docid: 11849
negative_passages.text: Size of a union of two sets
negative_passages.docid: 77480
negative_passages.text: Is the set $S=\left\{\left(z_1,z_2\right)\in \mathbb C\times \mathbb C:z_1^2+z_2^2=1\right\}.$ compact?