Coherent Space

We now define three new constructions. A set is down-closed if: (Def.1) for all a, b such that a ∈ it and b ⊆ a holds b ∈ it. A set is binary complete if: (Def.2) for every A such that A ⊆ it and for all a, b such that a ∈ A and b ∈ A holds a ∪ b ∈ it holds ⋃ A ∈ it. Let us observe that there exists a down-closed binary complete non-empty set. A coherent space is a down-closed binary complete non-empty set. In the sequel C, D are coherent spaces. Next we state four propositions: (1) ∅ ∈ C. (2) 2 is a coherent space. (3) {∅} is a coherent space. (4) If x ∈ ⋃ C, then {x} ∈ C. Let C be a coherent space. The functor Web(C) yields a tolerance of ⋃ C and is defined by: