semirings and basis of a topology Let $S$ be a semiring of subsets of a nonempty set $X$. What additional requirements must be satisfied for $S$ to be a base for a topology on $X$?
Prove that if such is the case, then each member of $S$ is both open and closed in this topology. What about all the open sets, are they closed too?
 A: There are several equivalent definitions of semiring of sets. I will use the definition at link 1
(I) It is sufficient that a semiring $S$  of subsets of a set $X$ to cover the set $X$  (i.e. $\bigcup S = X$) for $S$ to be base of a topological space on $X$.
Proof.
Suppose $S$ is such a semiring which covers  $X.\ $  
A necessary and sufficient condition for a set $B$ of subsets of a set $X$ to be a base of a topological space on $X$ is to satisfy two conditions:


*

*$B$ covers  $X.\ $

*Let $b_1, b_2$ be elements of $B$ and let $I$ be their intersection. Then for each $x$ in $I$, there is an element $b_3$ of $B$ containing $x$ and contained in $I$.
(check this link 2).
Since $S$ satisfies 1., it remains to prove that $S$ satisfies 2.
Let  $b_1, b_2$ be elements of $S$, $I$ be their intersection, and $x$ be in $I$.  Since $S$ is a semiring, it is $\cap-$`++`stable, and thus $I$ is in $S$, and 2. is satisfied.
(II). If the semiring $S$  of subsets of $X$ covers the set $X$, and thus, it is a base of a topology $T$ on $X$, then each member $s$ of $S$ is both close and open.
Proof.
Let  $s$ be member of $S. \ $ According the definition at link 1, (3'), $s$ is a union of a finite number of  disjoint members of $S$ (take  $A =s$, and $B = \emptyset$). Since $S$ is the base of topology $T$, this union of open sets is also an open set in $T$.
Now prove that $s$ is a closed set, i.e  ($X \setminus s$) is an open set in $T$. Use again link 1, (3'), now setting $A = X$ and $B = s$, and get that $X \setminus s$ is a union of a finite number of  disjoint members of $S$. Since $S$ is the base of topology $T$, this union of open sets is also an open set.
We got that in the topology generated by a semiring of subsets of a set,which covers this set,  all open sets are also closed and conversely (I did not know this). 
The sets of a topological space which ar both open and closed, are called clopen. Notice these facts:
. A topological space $X$ is discrete if and only if all of its subsets are clopen.
. Using the union and intersection as operations, the clopen subsets of a given topological space $X$ form a Boolean algebra. Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
You can find these facts at link 3.
