query: all $A_i$ are connected sets such that $\bigcap\limits_{i\in E} A_i \neq \emptyset$ then $\bigcup\limits_{i \in E} A_i$ is connected
positive_passages.docid: 158189
positive_passages.text: If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected
negative_passages.docid: 57943
negative_passages.text: If $G$ is a convex set, prove that $\mathbb{C}\cup\{\infty\}\setminus G$ is connected.
negative_passages.docid: 116866
negative_passages.text: $A\times B$ connected component implies $A,B$ connect components
negative_passages.docid: 91237
negative_passages.text: Every infinite set contains a countable subset
negative_passages.docid: 127919
negative_passages.text: prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected.
negative_passages.docid: 2575
negative_passages.text: Showing that $\mathbb{R}$ is connected
negative_passages.docid: 158850
negative_passages.text: Is it true that a boundary of a simply connected and bounded set is connected in $\mathbb{C}$?
negative_passages.docid: 36878
negative_passages.text: Prove that $\operatorname{SL}(n,\Bbb R)$ is connected.
negative_passages.docid: 99236
negative_passages.text: Interior of closure of an open set
negative_passages.docid: 112731
negative_passages.text: If $Y$ is connected, why is $A\cup Y$ connected in this case?
negative_passages.docid: 68918
negative_passages.text: Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$
negative_passages.docid: 28391
negative_passages.text: Prove the compactness of a set B
negative_passages.docid: 151719
negative_passages.text: Connected set in $\mathbb{S}^1$
negative_passages.docid: 153107
negative_passages.text: Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?
negative_passages.docid: 16654
negative_passages.text: Unions and Intersections of Open Sets are Open
negative_passages.docid: 84936
negative_passages.text: The set of all finite sequences of members of a countable set is also countable
negative_passages.docid: 39746
negative_passages.text: How to prove that given set is a connected subset of the space of matrices?