I very often have to write something like:
$\exists U,V\subseteq M$ where $U,V$ are open, but there's no short hand for it. On my written notes, I do tend to write something like:
$\exists U,V\mathop{\subseteq}_\text{open}M$ is there a common short hand though?
Example of a short hand that exists
$\newcommand{\bigudot}{\mathop{\bigcup\mkern-14mu\cdot\mkern5mu}}$
$\newcommand{\udot}{\cup\mkern-11.5mu\cdot\mkern5mu}$
It is established already that $\bigudot$ means "union of sets that are pairwise disjoint", so if I write:
$\forall A\exists\mathcal{A}:A\subseteq\bigudot\mathcal{A}$ – or something – it is clear from the context that $\mathcal{A}$ is a family of pairwise disjoint sets
So my question is this:
Is there a notation for this already, like perhaps a $\subset$ with a dot in to mean "open subset"
Best Answer
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, $$ \forall U\in \mathcal{T}_Y, ~ f^{-1}(U) \in \mathcal{T}_X $$