What is the symbol $\triangleq$? I came across this new symbol while reading a document about writing proofs, and I have never seen it before.
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 $U_x$.
It actually is defined as "estimates", and also means that the content on the two sides correlates or conforms to each other.

For further reference, see here and here, the symbol is quite often used in (German?) academic context.