I have this problem:
"Show that a topological space X$\neq\emptyset$ is irreducible $\Leftrightarrow$ $\forall$ U open subset of X, U is connected. "
I can easily prove the $\Rightarrow$ part, using that every open subset of an irreducible space is itself irreducible (therefore connected).
But what about the $\Leftarrow$ part? I have no idea how to prove it, and to be fair I'm not even sure it's true.