A Topological Approach to Chemical Organizations-SUPPLEMENT

This supplement extends Section 7 on the main text, providing additional material on generalized topologies and pertinent references. 1 Further Remarks on Generalized Topologies This text summarizes, and in part slightly extends, results by Day [5], Hammer [7, 11] and Gni lka [8] on structures defined by a set X endowed with an arbitrary set-valued set-function. Let c′ and c′′ be two generalized closure operators on X. We say that c′ is finer than c′′, c′ c′′, or c′′ is coarser than c′ if c′(A) ⊆ c′′(A) for all A ∈ P(X). Note that c′ c′′ and c′ c′′ implies c′ = c′′. A function f : (X, cl) → (Y, cl) is closure preserving if for all A ∈ P(X), f(cl(A)) ⊆ cl(f(A)) holds; continuous if for all B ∈ P(Y ), cl(f−1(B)) ⊆ f−1(cl(B)) holds. It is obvious that the identity ı : (X, cl) → (X, cl) : x 7→ x is both closure-preserving and continuous since ı(cl(A)) = cl(A) ⊆ cl(A) = cl(ı(A)). Furthermore, the concatenation h = g(f) of the closurepreserving (continuous) functions f : X → Y and g : Y → Z is again closure-preserving (continuous). Let (X, cl) and (Y, cl) be two sets with arbitrary closure functions and let f : X → Y . Then the following conditions (for continuity) are equivalent, see e.g. [9, Thm.3.1.]: (i) cl(f−1(B)) ⊆ f−1(cl(B)) for all B ∈ P(Y ). Manuscript 6 November 2008 Table 1 Basic axioms for Generalized Topologies. The properties below are meant to hold for all A,B ∈ P(X) and all x ∈ X, respectively. closure interior neighborhood K0’ ∃A : x / ∈ cl(A) ∃A : x ∈ int(A) N (x) 6= ∅ K0 cl(∅) = ∅ int(X) = X X ∈ N (x) K1 A ⊆ B =⇒ cl(A) ⊆ cl(B) A ⊆ B =⇒ int(A) ⊆ int(B) N ∈ N (x) and N⊆N ′ isotonic, cl(A ∩B) ⊆ cl(A) ∩ cl(B) int(A) ∪ int(B) ⊆ int(A ∪B) =⇒ monotone cl(A) ∪ cl(B) ⊆ cl(A ∪B) int(A ∩B) ⊆ int(A) ∩ int(B) N ′ ∈ N (x) KA cl(X) = X int(∅) = ∅ ∅ / ∈ N (x) KB A ∪B = X =⇒ A ∩B = ∅ =⇒ N ′, N ′′ ∈ N (x) =⇒ cl(A) ∪ cl(B) = X int(A) ∩ int(B) = ∅ N ′ ∩N ′′ 6= ∅ K2 A ⊆ cl(A) int(A) ⊆ A N ∈ N (x) =⇒ x ∈ N expansive K3 cl(A ∪B) ⊆ cl(A) ∪ cl(B) int(A) ∩ int(B) ⊆ int(A ∪B) N ′, N ′′ ∈ N (x) =⇒ sub-linear N ′ ∩N ′′ ∈ N (x) K4 cl(cl(A)) = cl(A) int(int(A)) = int(A) N ∈ N (x) ⇐⇒ idempotent int(N) ∈ N (x) K5 N (x) = ∅ or ∃N(x) :