query: If $Y_1,...,Y_n$ are a finite collection of compact subsets of $X$, then their union $Y_1\cup....\cup Y_n$ are also compact.
positive_passages.docid: 147213
positive_passages.text: Prove that finite unions of compact sets are compact
negative_passages.docid: 146664
negative_passages.text: The union of two linearly independent subsets
negative_passages.docid: 67259
negative_passages.text: Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
negative_passages.docid: 144360
negative_passages.text: For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
negative_passages.docid: 64694
negative_passages.text: $C_{c}(X)$ is complete. then implies that $X$ is compact.
negative_passages.docid: 61930
negative_passages.text: Intersection of Closed and Compact Set is Closed
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: 12404
negative_passages.text: Union of definable sets
negative_passages.docid: 14219
negative_passages.text: Is the intersection of a closed set and a compact set always compact?
negative_passages.docid: 93419
negative_passages.text: Let $\mathcal A$ be a collection of pairwise disjoint subsets of a $\sigma$-finite measure space,Show that $\mathcal A$ is at most countable.
negative_passages.docid: 11383
negative_passages.text: Disjoint compact sets in a Hausdorff space can be separated
negative_passages.docid: 136094
negative_passages.text: $G_\delta$ singletons in compact Hausdorff and first countability
negative_passages.docid: 158498
negative_passages.text: $A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
negative_passages.docid: 10531
negative_passages.text: Is the convex hull of a compact set compact?
negative_passages.docid: 80096
negative_passages.text: Prove that the union of countably many countable sets is countable.
negative_passages.docid: 22854
negative_passages.text: Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B$ closed?
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