regular open algebra
From the parent entry, all of the operations above are well-defined (that the result sets are regular open). Also, we have the following:
is a Boolean algebra
We break down the proof into steps:
is a lattice. This amounts to verifying various laws on the operations:
(idempotency of and ): Clearly, . Also, , since is regular open.
Commutativity of the binary operations are obvious.
The associativity of is also obvious. The associativity of goes as follows: , since is open (which implies that is regular open). The last expression is equal to . Interchanging the roles of and , we obtain the equation , which is just , or . The commutativity of completes the proof of the associativity of .
Finally, we verify the absorption laws. First, . Second, .
is complemented. First, it is easy to see that and are the bottom and top elements of . Furthermore, for any , . Finally, .
If LHS, then and for any open set with , we have that . In particular, is such an open set (for ), so that , or . Since is arbitrary, RHS. Now, apply the set complement, we have . Applying next we get for the LHS, and for RHS, since is closed. As reverses order, the new inclusion is
From this, a direct calculation shows , noticing that the first and second inclusions use above (and the fact that preserves order), and the last equation uses the fact that for any open set , is regular open. This proves the .
Finally, to finish the proof, we only need to show one of two distributive laws, say, , for the other one follows from the use of the distributive inequalities. This we do be direct computation: .
Clearly, every clopen set is regular open. In addition, . If is clopen, so is the complement of its closure, and hence . If are clopen, so is their intersection . Similarly, is clopen, so that is clopen also. ∎
In fact, is a complete Boolean algebra. For an arbitrary subset of , the meet and join of are and respectively.
is the smallest complete Boolean subalgebra of extending .
More to come…
|Title||regular open algebra|
|Date of creation||2013-03-22 17:56:21|
|Last modified on||2013-03-22 17:56:21|
|Last modified by||CWoo (3771)|