What is the symbol $\triangleq$?
It’s is defined to be equal to; it’s my preferred symbol, but the most common one is $:=$, and I’ve also seen $\overset{\text{def}}=$.
For each $x\in X$ there is an open nbhd $U_x$ of $x$ such that ... . Then $\mathscr{U}\triangleq\{U_x:x\in X\}$ is ...
The $\triangleq$ indicates that $\mathscr{U}$ is being defined to be $\{U_x:x\in X\}$: we are not saying that some previously defined $\mathscr{U}$ is equal to the collection of these sets $U_x$.
The symbol means it is "equal to by definition".