Is there a notation for "open subset"
Remember that a topological space $X$ is really a pair $(X,\mathcal{T})$ where $\mathcal{T}$ is a collection of "open" subsets of $X$. We often drop this cumbersome notation. But if you write, $U\in \mathcal{T}$, then it is clear you are talking about open sets.
For example, you can define continuity $f:X\to Y$ by saying, $$ \forall U\in \mathcal{T}_Y, ~ f^{-1}(U) \in \mathcal{T}_X $$