Foundation Arc Complement Acyclic Homeo Hom Comp Symm

The homeomorphism lemma

A homeomorphism is a continuous map between two topological spaces that has a continuous inverse. The declaration homeoHom_comp_symm in the framework's machine-checked library of formal theorems states a basic property: if you take a homeomorphism, follow it by its inverse, you get the identity map on the original space. In symbols, for a homeomorphism e from space A to space B, the composition homeoHom e followed by homeoHom e.symm equals the identity on A. This is a theorem, proved in the Lean 4 proof assistant, and it is a direct consequence of the definition of a homeomorphism's inverse.

The lemma is a small piece of a larger argument about the complements of arcs in high-dimensional spheres. The framework uses it to show that a certain map induced by a homeomorphism on a complement is also a homeomorphism, which is a step in proving that these complements have trivial homology. The lemma itself, however, does not establish any property about the specific arcs or spheres in that argument; it is a general statement about any homeomorphism between any two topological spaces.

What the lemma does not claim is equally precise. It does not say that every continuous bijection is a homeomorphism; that requires the inverse to be continuous as well. It does not claim that the composition of a homeomorphism with its inverse is the identity on the codomain; that is the companion lemma homeoHom_symm_comp, which states the composition in the other order equals the identity on B. The lemma also does not assert that the homeomorphism is an isometry or preserves any geometric structure; it only concerns the topological structure of the spaces.

In the context of the framework's forcing chain, this lemma is a technical tool, not a main result. It is used to ensure that certain maps between homology groups are well-defined and that the algebraic structures behave as expected. Its proof is a simple calculation using the properties of the inverse map, and it shows the rigor of the library in formally verifying even such a basic fact.

lemma homeoHom_comp_symm {A B : Type} [TopologicalSpace A] [TopologicalSpace B]
    (e : A ≃ₜ B) : homeoHom e ≫ homeoHom e.symm = 𝟙 (TopCat.of A) := by
  ext x
  exact e.symm_apply_apply x

What this page does not claim

The lemma does not claim that every continuous bijection is a homeomorphism. The lemma does not claim that the composition of a homeomorphism with its inverse is the identity on the codomain. The lemma does not assert that the homeomorphism preserves any geometric structure beyond topology.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/ArcComplementAcyclic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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