Let $X$ be a topological space and let $\mathscr{F}$ be the smallest collection of subsets of $X$ which contains all open subsets of $X$ and is closed with respect to the formation of finite intersections and complements.
I solved Ex7.20(i), which says that a subset $E$ of $X$ belongs to $\mathscr{F}$ iff $E$ is a finite union of sets of the form $U \cap C$, where $U$ is open and $C$ is closed.
What is the name of $\mathscr{F}$? I found that if $X$ is a notherian topological space, it is called Constructible sets. Is this name used for any $X$, or is there no special name for the case of non-noetherian space $X$?
Is it true that any finite union $\cup_{i=1}^n (U_i \cap C_i)$(where $U_i$ is open and $C_i$ is closed) can be presented as $U \cap C$? One of the solution of the problem states this, but I doubt it is true. I tried to find a counterexample in $\mathbb{R}^2$ by sketching some sets, but I couldn't find a one.