Finite disjoint union of the proper differences of the compact set forms a ring It would be great if someone can give a proof that in a locally compact Hausdorff space, the class of sets that are all sets which are the finite disjoint unions of the proper difference of the compact sets forms a ring.
EDIT
To clarify, the ring discussed here is the ring of sets defined as
sets that are closed under operations

*

*difference (i.e., relative compliment)

*union

that is,
if $A,B \in R$

*

*$A-B \in R$

*$A \cup B  \in R$
finite disjoint union of the Proper difference of the compact set means
$$\cup_n E_n$$
where each $E_n$ is disjoint sets ranging from n=1,...,N and
$$E_n=(A_n-B_n)$$
where it is assumed that $B_n \subset A_n$
$ A_n$ and $B_n\in C$
and $C$ is a class of compact sets in a locally compact Hausdorff space.
 A: Let $X$ be a locally compact Hausdorff space and $\mathcal A$ be the family of subsets of $X$ which are the finite unions of  pairwise disjoint differences of  compact subsets of $X$. Let us check that $\mathcal A$ is a ring.
Let $E,F$ be any members of $\mathcal A$. There exist finite families $\{E_i\}_{i=1}^n$ and $\{F_j\}_{j=1}^m$ of pairwise disjoint subsets of $X$ such that for each $i$ and $j$ there exist compact subsets $A_i\supset B_i$ and $C_j\supset D_j$ such that $E_i=A_i-B_i$ and $F_j=C_j- D_j$.
We claim that $E-F\in\mathcal A$. Since $\mathcal A$ is closed with respect to unions of disjoint sets and $E-F$ is a union of a family $\{E_i – F\}$ of disjoint sets,  it suffices to show that $E_i –F\in\mathcal A$ for each $i$.
But $$E_i –F=E_i-(\bigcup_j C_j- D_j)=\bigcap_j (E_i-(C_j-D_j))= \bigcap_j (E_i\cap D_j)\cup (E_i-C_j).$$
The latter intersection is a finite union of disjoint sets which are intersections of the form $\bigcap_j G_j$, where each $G_j$ is either $E_i\cap D_j$ or $E_i-C_j$. That is there exists a subset $S$ of $\{1,\dots m\}$ such that
$$\bigcap_j G_j= E_j\cap \bigcap_{j\in S} D_j \cap \bigcap_{j\in \{1,\dots m\}-S} (E_i - C_j)=E_j\cap \bigcap_{j\in S} D_j \cap E_j - \bigcup_{j\in \{1,\dots m\}-S} C_j\in\mathcal A$$ (here, for the convenience, we assume that the intersection of an empty family equals $X$).
For each $i\in\{1,\dots,n\}$ and $j\in\{1,\dots,m\}$ put $G_{ij}=E_i\cap F_j=(A_i\cap C_j)-(B_i\cap D_j)\in\mathcal A$.
Then $E\cap F=\bigcup_{ij} G_{ij}$ and $$E\cup F=(E\setminus F)\cup (F\setminus E)\cup (E\cap F)= (E\setminus F)\cup (F\setminus E)\cup \bigcup_{ij} G_{ij}.$$ Since the sets $E_i-\bigcup_j F_j$, $F_j-\bigcup_i E_j$, and $G_{ij}$ are pairwise disjoint, we see that $E\cup F\in\mathcal A$.
