Please is this prof is correct ? Let $\{\Omega_i\}_{i\in I}$ a famille of connected sets such that $$\forall i,j\in I, \Omega_i\cap\Omega_j\neq\emptyset$$ I want to prove that $\bigcup_{i\in I} \Omega_i$ is connected.
If I suppose that $\bigcup \Omega_i$ is not connected then, there exists two non empty open sets $A,B$ from $\bigcup \Omega_i$ such that $$ \begin{cases} \bigcup \Omega_i= A\cup B\\ A\cap B=\emptyset \end{cases} $$ we have $\forall i\in I, \Omega_i\subset \bigcup_{i\in I}\Omega_i=A\cup B$ then by the connectedness of $\Omega_i$ $$\forall i\in I, [\Omega_i\subset A ~\text{or}~ \Omega_i\subset B]$$
As $\forall i,j\in I, A_i\cap A_j\neq \emptyset$ we deduce that $$\forall I\in A, \Omega_i\subset A~\text{or}~ \forall i\in I,\Omega_i\subset B$$ it follows that $B=\emptyset$ or $D=\emptyset$, which is a contradiction.
Please if I change the condition $$\forall i,j\in I, \Omega_i\cap\Omega_j\neq\emptyset$$ by $$ \exists i_0\in I, \Omega_{i_0}\cap \Omega_j\neq \emptyset,\forall j\in I $$ How to do ? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\{\Omega_i\}_{i\in I}$ a famille of connected sets such that $$\exists i_0\in I, \Omega_{i_0}\cap \Omega_j\neq \emptyset,\forall j\in I$$ I want to prove that $\bigcup_{i\in I} \Omega_i$ is connected.
If I suppose that $\bigcup \Omega_i$ is not connected then, there exists two non empty open sets $A,B$ from $\bigcup \Omega_i$ such that $$ \begin{cases} \bigcup \Omega_i= A\cup B\\ A\cap B=\emptyset \end{cases} $$ we have $\forall i\in I, \Omega_i\subset \bigcup_{i\in I}\Omega_i=A\cup B$ then $\Omega_{i_0}\subset A\cup B$ as it is connected we have $$\Omega_{i_0}\subset A~\text{or} ~ \Omega_{i_0}\subset B$$ if we suppose that $\Omega_{i_0}\subset A$ then $$\forall j\in I, \Omega_{j}\cap A\neq \emptyset $$ by the connectedness of $\Omega_i$ we deduce that $$\forall j\in I, \Omega_{j}\cap B=\emptyset$$ then $$\forall j\in I, \Omega_j\subset A$$ thus $$\bigcup_{j\in I}\Omega_j\subset A$$ so $B=\emptyset$ contradiction in the same way if we suppose that $\Omega_{i_0}\subset B$ we find that $A=\emptyset$
thank you