regular open algebra


A regular open algebra is an algebraic system π’œ whose universePlanetmathPlanetmath is the set of all regular open sets in a topological spaceMathworldPlanetmath X, and whose operationsMathworldPlanetmath are given by

  1. 1.

    a constant 1 such that 1:=X,

  2. 2.

    a unary operation β€² such that for any U,  Uβ€²:=UβŠ₯, where UβŠ₯ is the complementPlanetmathPlanetmath of the closureMathworldPlanetmathPlanetmath of U in X,

  3. 3.

    a binary operationMathworldPlanetmath ∧ such that for any U,Vβˆˆπ’œ, U∧V:=U∩V, and

  4. 4.

    a binary operation ∨ such that for any U,Vβˆˆπ’œ, U∨V:=(UβˆͺV)βŠ₯βŠ₯.

From the parent entry, all of the operations above are well-defined (that the result sets are regular open). Also, we have the following:

Theorem 1.

π’œ is a Boolean algebraMathworldPlanetmath

Proof.

We break down the proof into steps:

  1. 1.

    π’œ is a latticeMathworldPlanetmath. This amounts to verifying various laws on the operations:

    • –

      (idempotency of ∨ and ∧): Clearly, U∧U=U. Also, U∨U=(UβˆͺU)βŠ₯βŠ₯=UβŠ₯βŠ₯=U, since U is regular open.

    • –

      Commutativity of the binary operations are obvious.

    • –

      The associativity of ∧ is also obvious. The associativity of ∨ goes as follows: U∨(V∨W)=(Uβˆͺ(VβˆͺW)βŠ₯βŠ₯)βŠ₯βŠ₯=UβŠ₯∩(VβˆͺW)βŠ₯⁣βŠ₯βŠ₯=UβŠ₯∩(VβˆͺW)βŠ₯, since VβˆͺW is open (which implies that (VβˆͺW)βŠ₯ is regular open). The last expression is equal to UβŠ₯∩(VβŠ₯∩WβŠ₯). Interchanging the roles of U and W, we obtain the equation W∨(V∨U)=WβŠ₯∩(VβŠ₯∩UβŠ₯), which is just UβŠ₯∩(VβŠ₯∩WβŠ₯), or U∨(V∨W). The commutativity of ∨ completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof of the associativity of ∨.

    • –

      Finally, we verify the absorption laws. First, U∧(U∨V)=U∩(UβˆͺV)βŠ₯βŠ₯=UβŠ₯βŠ₯∩(UβˆͺV)βŠ₯βŠ₯=(UβŠ₯βˆͺ(UβˆͺV)βŠ₯)βŠ₯=(UβŠ₯βˆͺ(UβŠ₯∩VβŠ₯)βŠ₯=(UβŠ₯)βŠ₯=U. Second, U∨(U∧V)=(Uβˆͺ(U∨W)βŠ₯βŠ₯=UβŠ₯βŠ₯=U.

  2. 2.

    π’œ is complemented. First, it is easy to see that βˆ… and X are the bottom and top elements of π’œ. Furthermore, for any Uβˆˆπ’œ, U∧Uβ€²=U∩UβŠ₯=U∩(Xβˆ–UΒ―)βŠ†U¯∩(Xβˆ–UΒ―)=βˆ…. Finally, U∨Uβ€²=(UβˆͺUβŠ₯)βŠ₯βŠ₯=(UβŠ₯∩UβŠ₯βŠ₯)βŠ₯=(UβŠ₯∩U)βŠ₯=βˆ…βŠ₯=X.

  3. 3.

    π’œ is distributive. This can be easily proved once we show the following: for any open sets U,V:

    (*)  UβŠ₯βŠ₯∩VβŠ₯βŠ₯=(U∩V)βŠ₯βŠ₯.

    To begin, note that since U∩VβŠ†U, and βŠ₯ is order reversing, (U∩V)βŠ₯βŠ₯βŠ†UβŠ₯βŠ₯ by applying βŠ₯ twice. Do the same with V and take the intersectionMathworldPlanetmath, we get one of the inclusions: (U∩V)βŠ₯βŠ₯βŠ†UβŠ₯βŠ₯∩VβŠ₯βŠ₯. For the other inclusion, we first observe that

    U∩VΒ―βŠ†U∩VΒ―.

    If x∈ LHS, then x∈U and for any open set W with x∈W, we have that W∩Vβ‰ βˆ…. In particular, U∩W is such an open set (for x∈U∩W), so that (U∩W)∩Vβ‰ βˆ…, or W∩(U∩V)β‰ βˆ…. Since W is arbitrary, x∈ RHS. Now, apply the set complement, we have (U∩V)βŠ₯βŠ†U∁βˆͺVβŠ₯. Applying βŠ₯ next we get (U∩V)βŠ₯βŠ₯ for the LHS, and (U∁βˆͺVβŠ₯)βŠ₯=U∁-∁∩VβŠ₯βŠ₯=U∁⁒∁∩VβŠ₯βŠ₯=U∩VβŠ₯βŠ₯ for RHS, since U∁ is closed. As βŠ₯ reverses order, the new inclusion is

    (**)  U∩VβŠ₯βŠ₯βŠ†(U∩V)βŠ₯βŠ₯.

    From this, a direct calculation shows UβŠ₯βŠ₯∩VβŠ₯βŠ₯βŠ†(UβŠ₯βŠ₯∩V)βŠ₯βŠ₯βŠ†(U∩V)βŠ₯⁣βŠ₯⁣βŠ₯βŠ₯=(U∩V)βŠ₯βŠ₯, noticing that the first and second inclusions use (**) above (and the fact that βŠ₯βŠ₯ preserves order), and the last equation uses the fact that for any open set W, WβŠ₯ is regular open. This proves the (*).

    Finally, to finish the proof, we only need to show one of two distributive laws, say, U∧(V∨W)=(U∧V)∨(U∧W), for the other one follows from the use of the distributive inequalities. This we do be direct computation: U∧(V∨W)=U∩(VβˆͺW)βŠ₯βŠ₯=UβŠ₯βŠ₯∩(VβˆͺW)βŠ₯βŠ₯=(U∩(VβˆͺW))βŠ₯βŠ₯=((U∩V)βˆͺ(U∩W))βŠ₯βŠ₯=((U∧V)βˆͺ(U∧W))βŠ₯βŠ₯=(U∧V)∨(U∧W).

Since a complemented distributive latticeMathworldPlanetmath is Boolean, the proof is complete. ∎

Theorem 2.

The subset B of all clopen sets in X forms a Boolean subalgebra of A.

Proof.

Clearly, every clopen set is regular open. In addition, 1βˆˆβ„¬. If U is clopen, so is the complement of its closure, and hence Uβ€²βˆˆβ„¬. If U,V are clopen, so is their intersection U∧V. Similarly, UβˆͺV is clopen, so that U∨V=UβˆͺV is clopen also. ∎

Theorem 3.

In fact, A is a complete Boolean algebra. For an arbitrary subset K of A, the meet and join of K are (β‹‚{U∣U∈K})βŠ₯βŠ₯ and (⋃{U∣U∈K})βŠ₯βŠ₯ respectively.

Proof.

Let V=(⋃{U∣Uβˆˆπ’¦})βŠ₯βŠ₯. For any Uβˆˆπ’¦, UβŠ†β‹ƒ{U∣Uβˆˆπ’¦} so that U=UβŠ₯βŠ₯=(⋃{U∣Uβˆˆπ’¦})βŠ₯βŠ₯=V. This shows that V is an upper bound of elements of 𝒦. If W is another such upper bound, then UβŠ†W, so that ⋃{U∣Uβˆˆπ’¦}βŠ†W, whence V=(⋃{U∣Uβˆˆπ’¦})βŠ₯βŠ₯βŠ†WβŠ₯βŠ₯=W. The infimumMathworldPlanetmathPlanetmath is proved similarly. ∎

Theorem 4.

π’œ is the smallest complete Boolean subalgebra of P⁒(X) extending B.

More to come…

Title regular open algebra
Canonical name RegularOpenAlgebra
Date of creation 2013-03-22 17:56:21
Last modified on 2013-03-22 17:56:21
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 16
Author CWoo (3771)
Entry type Definition
Classification msc 06E99