Let $(X,\tau)$ be connected such that $\emptyset$ and $X$ are the only open connected subsets. Does this imply that $\tau = \{\emptyset, X\}$?



Connected topological space $X$ such that $\emptyset, X$ are the only open connected subsets
Let $X\_0$ be any topological space in which no nonempty open set is connected, e.g., $X\_0 = \mathbb{Q}$ with the usual topology. Then let $X = X\_0 \cup \{\infty\}$ with open sets $\{$all open subsets of $X\_0\} \cup \{X\}$. The whole space is connected because the only open set containing $\infty$ is the whole space, and any proper nonempty open subset cannot contain $\infty$ and hence must be disconnected by the assumption on $X\_0$.


