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34 Chapter 6. Some special sublocales
6.2.6. Corollary. We have
(1) c(b)⊆o(a) iff a∨b=1, (2) o(b)⊆c(a) iff a∧b=0.
Proof. (1) c(b) ⊆ o(a) iff c(b) ∩ c(a) = O iff c(a ∨ b) = O iff a ∨ b = 1.
(2) o(b) ⊆ c(a) iff o(b) ∩ o(a) = O iff o(a ∧ b) = O iff a ∧ b = 0.
Let us now turn to the naturally defined
6.3. Closure of a sublocale S. It is the sublocale
Obviously
S =↑ S = c( S).
S is the least closed sublocale containing S.
Proposition. We have O = O, S = S and S ∨ T = S ∨ T .
Proof. The first two equations are trivial, and the third one is very easy: set
a = S, b = T. Then
S∨T = ↑a∨↑b
= {x∧y|x≥a,y≥b}
= ↑(a∧b)
= ↑ (S∨T) = S∨T.
A sublocale S is dense if S = L. Obviously, then, S is dense iff 0 ∈ S.

6.4. Density.
O n t h e ot h e r h a n d , i f 0 ∈ S t h e n n e c e s s a r i l y { x → 0 | x ∈ L } ⊆ S . B u t b y the formula ( -distr) in 1.3
BL ={x→0|x∈L}(={x∈L|x∗∗ =x} where x∗ =x→0)
is itself a sublocale and, hence, the smallest dense sublocale. Thus we have
obtained the Isbell’s Density Theorem
Proposition. BL is the least dense sublocale of L.