Proposition 2

The union of a family of connected sets whose intersection is non-empty is connected.

Let $(A_i)_{i \in I}$ be a family of connected subsets of $X$, all of which contain the same point $x$; we have to show that

is connected. If not, there are two open sets $B$ and $C$ such that $B \cap A$ and $C \cap A$ are non-empty, and $A \subset B \cup C$ and $A \cap B \cap C = \emptyset$. $x$ belongs to one of the sets $B, C$, say $x \in B$; on the other hand one of the sets $A_i$, say $A_x$, meets $C$; we have therefore $A_x \subset B \cup C$, $A_x \cap B \cap C = \emptyset$ and $B \cap A_x$ and $C \cap A_x$ are non-empty. Hence $A_x$ is not connected, which is a contradiction.