$\forall \beta \in I$, $A_{\beta }$ is connected, and $\left ( \bigcup_{\alpha < \beta }A_{\alpha } \right )\cap A_{\beta }\neq \varnothing$ . Is $\bigcup_{\alpha \in I}A_{\alpha } $connected? For the index set $I$ , when it is countable ,the answer is obvious. I want to know the general conclusion. Thanks！