Is it true that openness is transitive?
Its clear for me in metric spaces. But how about topological spaces?
To make it clear:
Let $X$ be a topological space. Then it holds: If $U$ is open in $X$ and $V\subseteq U$ open in $U$ then $V$ is open in $X$. Of course we consider subspaces with the subset-topology.
Is this true? If yes. How it can be proven?