I am having difficulty understanding the notation for a homework problem (which is given directly as handout). I will reproduce the statement for the part of the question here.
$ \begin{array}{l}{\text { Let } M \text { be a regular surface. Deduce that if } x: U \rightarrow M \subset \mathbb{R}^{3} \text { is a regular chart }} \\ {\text { (possibly without a continuous inverse), then } x(A) \text { is open in } M \text { for every open }} \\ {\text { set } A \subset U . \text { Hint: decompose } x(A)=\cup_{y} x\left(A \cap W_{y}\right) \text { where } y: W_{y} \rightarrow M \text { is a }} \\ {\text { regular chart with continuous inverse. }}\end{array} $
I just do not understand what $\cup_{y}$ denotes (I would normally expect a union symbol to be subscripted over some index), it seems to me like extending the original function, but what exactly is it?
I'm pretty sure this is an indexed union, as you suspect. The hint is asking you to provide a family of regular charts $\{y_i : W_i \to M_i \subseteq M\}_{i \in I}$, each with a continuous inverse, such that $\bigcup_{i \in I} W_i = U$. Then $x(A) = \bigcup_{i \in I} x(A \cap W_i)$. The notation is perhaps a bit confusing but I don't what else the hint could mean.