Understanding Semialgebras: Exploring the Definition and Examples

[LeonhardEuler]

Probably a stupid question here, but I've been beating myself up over it and can't find a resolution. I'm reading a book (Probability: Theory and Examples by Rick Durrett) that defines a semialgebra as:

A collection of sets S is said to be a semialgebra if (i) it is closed under intersection, and (ii) if S is an element of S, then S^c is a finite disjoint union of sets in S.

Already, this seems extremely odd to me because the compliment of a set belongs to the same space as the set itself. The disjoint union introduces another index to each element, if I am understanding that correctly as according to http://mathworld.wolfram.com/DisjointUnion.html" . So unless we are dealing with strange sets that include elements of different dimensions, I don't see how this is possible. The book then goes on to show that I clearly have misunderstood something because it then gives an example of a semialgebra:

An important example of a semialgebra is R^d_o = the collection of sets of the form

(a₁,b₁]X ... X(a_d,b_d] , a subset of R^d where $$-\infty \leq a_{i} < b_{i} \leq \infty$$

But if i look at the interval (0,1] in R, then its compliment is
$$(-\infty,0] \cup (1,\infty)$$
Which is a union of intervals of the real line, not a disjoint union. A disjoint union would seem to have sets of the form $$(-\infty,0] \times {{0}}$$ which don't belong to the real line at all.

Where am I going wrong?

 

[MathematicalPhysicist]

A disjoint union is a union of disjoint sets, so what you wrote is indeed a disjoint union of two intervals.

 

[LeonhardEuler]

What about what it says in those two links? I have not seen that definition of a disjoint union.

 

[g_edgar]

What about what it says in those two links? I have not seen that definition of a disjoint union.

Ignore that Wikipedia page. It is irrelevant for this problem. In this problem, a "disjoint union" of sets is a union of sets where the sets are disjoint.

 

[LeonhardEuler]

Ignore that Wikipedia page. It is irrelevant for this problem. In this problem, a "disjoint union" of sets is a union of sets where the sets are disjoint.

Well that makes a lot more sense then. The last part that confuses me is that I can only write the compliment of that set with an interval that extends to +infinity which is right open, not right closed. How is this problem resolved.