Show that $A$ is connected given that $A\cup B$ and $A\cap B$ are connected
114599
$A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected
157218
If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$
158189
If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected
76610
Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
62911
Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected
43310
Proving $A\cap(B-C)$ is equal to $(A\cap B)-(A\cap C)$
64586
How to prove that $\neg((A\cap B)\cup (\neg A \cap C)) = (A\cap\neg B)\cup (\neg A\cap \neg C)$
82578
Show that if $G$ is $2-$connected, then for any two vertices $u$ and $v$ there exists a cycle $C$ such that $u, v \in V (C)$
115606
Proofs involving $ (A\setminus B) \cup (A \cap B)$
68258
Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$
109682
Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
103441
Proving $(A\cup B)\cap(B\cup C)\cap(C\cup A)=(A\cap B)\cup (A\cap C)\cup (B\cap C)$?
75604
Proof that $\partial(A\times B) = ((\partial A)\times B)\cup (A\times(\partial B))$
63442
Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected
87327
Unions and intersections: $(A \cup B = A ∪ C) \land (A \cap B = A ∩ C) \implies B = C.$
70538
Intersection and union of simply connected domain