Show that if the intersection is non-empty, then the union is connected of the following
23225
Union of connected subsets is connected if intersection is nonempty
84609
Unary intersection of the empty set
146364
$X$ is connected iff $\forall A\subset X,$ $\partial A\neq\emptyset$
14215
Show that the union $\bigcup A_n$ has empty interior in $X$
152193
The intersection of a closure with an open set
70538
Intersection and union of simply connected domain
1453
Prove that the graph is connected
87327
Unions and intersections: $(A \cup B = A ∪ C) \land (A \cap B = A ∩ C) \implies B = C.$
10127
Example of a sequence of closed and connected sets whose intersection is disconnected
39746
How to prove that given set is a connected subset of the space of matrices?
5132
Given $A\cap \overline{B}\neq \emptyset$, prove $A\cup B$ connected.
13132
If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$)
17363
Prove that the intersection of two equivalence relations is an equivalence relation.
123742
Proving connectedness of the $n$-sphere
114599
$A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected
114786
Question about Infimum. (conclusion about intersection)
155999
How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$