Topological Semantics

Origin. Tarski and McKinsey (1944). Modal logic via topology. Interior operator = necessity. Open sets = possible. Foundation for spatial modal logic.

Models. Topological spaces. Interior for □, closure for ◇. Opens give truth. S4 = all topological spaces.

Formalism.

Topological space: (X, τ) where τ = open sets. Closed under ∪, finite ∩. ∅, X ∈ τ.

Interior operator: int(A) = largest open subset of A. □φ true at x iff x ∈ int(⟦φ⟧).

Closure operator: cl(A) = smallest closed superset. ◇φ true at x iff x ∈ cl(⟦φ⟧).

Kuratowski axioms: int(A ∩ B) = int(A) ∩ int(B) int(A) ⊆ A int(int(A)) = int(A) int(X) = X

S4 correspondence: □φ → φ (T): int(A) ⊆ A □φ → □□φ (4): int(int(A)) = int(A) S4 complete for topological semantics.

S5 and discrete: S5 = indiscrete topology (only ∅, X open). Every set is clopen. □φ = X or ∅.

Alexandroff spaces: Every intersection of opens is open. Corresponds to preorders. Kripke frames special case.

Derived set: d(A) = limit points of A. Modal logic of derivative. Different from interior.

Symbols.

Metatheory. S4 = topological spaces. Completeness for ℝ (real line). Spatial logics extend. Descriptive frames.

Applies to. Spatial reasoning. Mereotopology. Point-free topology. Modal logic foundations.

Limitations. Less intuitive than Kripke. Not all modalities topological. Spatial structures specific.

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