# Proving the family of constructible sets is a boolean algebra.

A constructible set is a finite union of locally closed sets. A locally closed set is the intersection of an open set and a closed set. I need to show that this is closed under finite unions and complementation to show it is a boolean algebra. The former is trivial to show but I am stuck on the latter. The complement of a union of locally closed sets is the intersection of sets which are a union of a closed set and an open set. I need to show that this is a union of locally closed sets, but I cannot see how this is true.

Indeed, we can write the complement of a finite union of locally closed sets as a finite intersection of sets which are a union of open and closed sets. Consider a single such union: $$U \bigcup F$$ where $$U$$ is open and $$F$$ closed. Suppose we are working with topological space $$(X,\tau )$$. Then $$U=U\bigcap X$$ is locally closed since $$U$$ is open and $$X$$ is closed. Equally, $$F=F \bigcap X$$ is locally closed since $$F$$ is closed and $$X$$ open. Hence $$U\bigcup F$$ is a finite union of locally closed sets: a constructible set.
We now claim that the intersection of two constructible sets is constructible. Suppose $$A$$ and $$B$$ are constructible, where $$A=\bigcup ^{n} _{i=1} C_i$$, and $$B = \bigcup ^{m} _{j=1} D_j$$, for locally closed $$C_i$$ and $$D_j$$. Then $$A\bigcap B = \bigcup _{1\le i \le n, 1\le j \le m}(C_i \bigcap D_j)$$, the elements of $$X$$ which are in some $$C_i$$ and some $$D_j$$.
Next, $$C_i$$ and $$D_j$$ are locally closed so let $$C_i=U_i\bigcap F_i$$ and $$D_j=V_j\bigcap G_j$$ where $$U_i$$ and $$V_j$$ open, $$F_i$$ and $$G_j$$ closed. Then $$C_i\bigcap D_j = (U_i\bigcap F_i)\bigcap (V_j \bigcap G_j)=(U_i\bigcap V_j)\bigcap (F_i \bigcap G_j)$$, where $$U_i\bigcap V_j$$ is open and $$F_i \bigcap G_j$$ is closed, so $$C_i\bigcap D_j$$ is locally closed.
Therefore $$A\bigcap B$$ is a finite union of locally closed sets: it's constructible.