Talk:Complete Heyting algebra

Morphisms of Frames

Something looks a little strang to me. Morphsims of Frames are just said to be (monotone) functions preserving finite meets and arbitrary joins. But isn't it required for them to also preserve bottoms and tops?

For instance, given:

- A = {a, b | a ≤ b}

- B = {α, β, γ, δ | α ≤ β ≤ γ ≤ δ}

two complete lattices (with ∧ = min and ∨ = max), there is a function:

φ : {a ↦ β ; b ↦ γ}

which meets the conditions of a morphism of Frames, but for which we do not have φ(⊥)=⊥ or φ(⊤)=⊤.

So, should a morphism of Frames preserve top and bottom? If not, I think it should be noted, as it is noted that morphisms of Frames do not necessarily preserves implication. Sedrikov (talk) —Preceding undated comment added 10:18, 15 December 2012 (UTC)

Ok, forget it. φ is not a morphism of Frames, as it does not preserve empty join (⊥) or meet (⊤). It is just that I had wrong definition of ⊤ and ⊥ in mind. Sedrikov (talk) —Preceding undated comment added 12:40, 15 December 2012 (UTC)

Locale-valued functions

What is known about locale-valued continuous functions on point-set topological spaces? I am interested in locale-valued generalization of an ultrametric, i.e. such metric on a point-set space, defined by open "balls", that two "balls" can intersect non-trivially (unlike spaces with real-valued ultrametric), but the intersection must also be a "ball". Incnis Mrsi (talk) 11:47, 22 December 2012 (UTC)