Lectures on Lax - Algebraic Methods in General Topology

(1) Ord: objects are (pre)ordered sets (= sets with a reflexive and transitive relation, no antisymmetry condition), with monotone maps. Formally: (X, a) with 1. a(x, x), 2. a(x, y) ∧ a(y, z) a(x, z), f : (X, a) − → (Y, b) with a(x, y) b(f (x), f (y)). (2) Met: objects are (generalized) metric spaces (=sets with a function a : X ×X − → [0, ∞] that is 0 on the diagonal and satisfies the triangle inequality), with contractions (=non-expansive maps). Formally: (X, a) with 1. 0 ≥ a(x, x), 2. a(x, y) + a(y, z) ≥ a(x, z), f : (X, a) − → (Y, b) with a(x, y) ≥ b(f (x), f (y)). (3) UMet: the full subcategory of Met containing all ultrametric spaces, for which (2) 2 is strengthened to 2. max{a(x, y), a(y, z)} ≥ a(x, z). (4) Top : topological spaces and continuous maps. In order to expose the analogy with (1), we describe topological spaces in terms of ultrafilter convergence, e.g. as sets X with a suitable relation a ⊆ βX ×X, with maps that preserve this relation; here βX is the set of all ultrafilters on X. Formally: (X, a) with 1. a(˙ x, x), 2. a(X, y) ∧ a(y, z) a(X, z), f : (X, a) − → (Y, b) with a(x, y) b(f [x], f (y)). where for x ∈ βX : x ∈ A # ⇐⇒ A ∈ x; for x ∈ βX, B ⊆ Y : B ∈ f [x] ⇐⇒ f −1 [B] ∈ x. We also used an extension of the relation a to a relation a ⊆ ββX × βX: a(X, y) ⇐⇒ ∀A ∈ X, B ∈ y ∃x ∈ A, y ∈ B : a(x, y). Motivation for the axioms as well as explanations for their equivalence with the usual presentation of topological spaces (in terms of open sets or neighborhood systems) will be provided in Lecture 2.