Chapter 29
Topology — Open and Closed Sets in Flow Space

Flow Algebra

flow "Topology — Open and Closed Sets in Flow Space"

# ---------- CONCEPT ----------
# A topology specifies which subsets are 'open' — the neighborhoods where flow can vary without leaving the set.
# Closed sets are complements of open sets — they contain their limit points (flow can't escape by convergence).
# Interior, closure, and boundary describe how coherence fills, seals, and edges a set.
# Continuity is preservation of openness (preimages of open sets are open) — flow respects neighborhood structure.

# ---------- STATES ----------
state TopologicalSystem(T):
  Space:
    description: "X — underlying carrier of points/events."
  Topology:
    description: "𝒯 ⊆ 𝒫(X) — collection of open sets satisfying union/finite-intersection axioms."
  OpenSet:
    description: "U ∈ 𝒯 — every point has a neighborhood contained in U (wiggle room of flow)."
  ClosedSet:
    description: "C = X \\ U for some U ∈ 𝒯 — contains all its limit points."
  Interior:
    description: "int(A) — largest open set inside A (freely moving region)."
  Closure:
    description: "cl(A) — smallest closed set containing A (A plus its limit points)."
  Boundary:
    description: "∂A = cl(A) \\ int(A) — the edge where flow meets its complement."
  Basis:
    description: "ℬ — family of opens whose unions generate 𝒯 (local building blocks)."
  Neighborhood:
    description: "N(x) — any open set containing x (local flow room)."
  ContinuousMap:
    description: "f: (X,𝒯_X)→(Y,𝒯_Y) s.t. f^{-1}(V) open in X for every open V in Y."

# ---------- PARAMETERS ----------
params:
  carrier: "X"
  topology: "𝒯"
  basis: "ℬ"
  subset: "A ⊆ X"
  map: "f: X→Y"
  closure_operator: "cl(·)"
  interior_operator: "int(·)"
  boundary_operator: "∂(·)"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "∅, X ∈ 𝒯."
  - "Arbitrary unions of opens are open; finite intersections of opens are open."
  - "Closed sets: complements of opens; arbitrary intersections of closed are closed; finite unions of closed are closed."
  - "int(A) ⊆ A ⊆ cl(A), and A is open ⇔ A = int(A); closed ⇔ A = cl(A)."
  - "∂A = cl(A) \\ int(A) = cl(A) ∩ cl(X\\A)."
  - "f continuous ⇔ preimage of every open is open ⇔ preimage of every closed is closed."
  - "Basis ℬ generates 𝒯: U open ⇔ ∀x∈U, ∃B∈ℬ with x∈B⊆U."

# ---------- TRANSFORMS ----------
transforms:
  GenerateTopology(ℬ):
    rule: "𝒯 = {⋃𝒱 | 𝒱 ⊆ ℬ} with finite-intersection property among basis elements."
  InteriorOf(A):
    rule: "int(A) = ⋃{U ∈ 𝒯 | U ⊆ A}."
  ClosureOf(A):
    rule: "cl(A) = ⋂{C closed | A ⊆ C} = A ∪ A' (A' = set of limit points)."
  BoundaryOf(A):
    rule: "∂A = cl(A) \\ int(A)."
  PreimageOpen(f,V):
    rule: "Compute f^{-1}(V); check openness in X."
  ContinuityTest(f):
    rule: "f is continuous iff ∀B∈ℬ_Y, f^{-1}(B) is open in X."
  SubspaceTopology(Y⊆X):
    rule: "𝒯_Y = {U∩Y | U ∈ 𝒯_X}."
  ProductTopology(X×Y):
    rule: "Basis = {U×V | U open in X, V open in Y}."
  QuotientTopology(q):
    rule: "U open in X/~ iff q^{-1}(U) open in X."

# ---------- MONITORS ----------
monitors:
  OpennessCheck(U):
    measure: "∀x∈U, ∃B∈ℬ with x∈B⊆U"
    output: "True ⇒ U open."
  ClosednessCheck(C):
    measure: "X\\C open?"
    output: "True ⇒ C closed."
  LimitPointDetector(A):
    meaning: "x is limit point if every neighborhood N(x) meets A\\{x}."
  BoundaryDetector(A,x):
    meaning: "x in ∂A iff every neighborhood intersects both A and X\\A."
  ContinuityMonitor(f):
    meaning: "All basis preimages open ⇒ f continuous."
  ConvergenceChecker:
    meaning: "In T1 spaces, x∈cl(A) iff ∃ sequence/net in A→x."

# ---------- EQUATIONS / RELATIONS ----------
equations:
  - "int(A) = A^∘"
  - "cl(A) = \\overline{A}"
  - "∂A = \\overline{A} \\setminus A^∘ = \\overline{A} ∩ \\overline{X\\A}"
  - "f \\text{ continuous } \\Leftrightarrow \\forall V∈𝒯_Y, f^{-1}(V)∈𝒯_X"
  - "\\text{Subspace: } 𝒯_Y = \\{U∩Y: U∈𝒯_X\\}"
  - "\\text{Product basis: } ℬ_{X×Y} = \\{U×V: U∈𝒯_X, V∈𝒯_Y\\}"

# ---------- NOTES ----------
notes:
  - "Openness models freedom of variation (local wiggle room); closedness models completion under limits."
  - "Interior/closure/boundary partition behavior: A = int(A) ⊔ ∂A ⊔ (A^c∩cl(A))."
  - "Continuity is neighborhood preservation — flows map coherent neighborhoods to coherent neighborhoods."
  - "Subspace/product/quotient constructions let complex spaces inherit or reshape coherence."
  - "Metric examples: with d, open balls B_r(x) form a basis; topological ideas generalize beyond metrics."
  - "In Flow Algebra terms: openness = local flow capacity; closure = limit-sealed coherence; boundary = interface where flows from A and its complement meet."