{
  "id": "continua-arcs-and-cut-points",
  "title": "Continua, arcs and cut points",
  "description": "Locally connected continua with complete proofs: connected open sets are path connected, every path contains an arc, first Čech cohomology of a compact metric space by finite covers, and the theorem that a Peano continuum of dimension at most one with vanishing first Čech cohomology is a dendrite and has cut points, which is the last step of OpenAI's proof (2026) of the Singer conjecture in dimension four outside the middle degree.",
  "units": [
    {
      "id": "paths-in-locally-connected-spaces",
      "title": "Paths in locally connected spaces",
      "reader": "paths-in-locally-connected-spaces.html",
      "source": "src/paths-in-locally-connected-spaces.md",
      "source_sha256": "f865e51110cecede38c3c8e28d1df4b319cc6038e33f06737abe519a0acca0f4",
      "status": "author_self_checked"
    },
    {
      "id": "every-path-contains-an-arc",
      "title": "Every path contains an arc",
      "reader": "every-path-contains-an-arc.html",
      "source": "src/every-path-contains-an-arc.md",
      "source_sha256": "a8b128d51949f45ec877c82e9521f8887f223fcd897a9a213eb8711a023d3abd",
      "status": "author_self_checked"
    },
    {
      "id": "first-cech-cohomology-of-a-compact-metric-space",
      "title": "First Čech cohomology of a compact metric space",
      "reader": "first-cech-cohomology-of-a-compact-metric-space.html",
      "source": "src/first-cech-cohomology-of-a-compact-metric-space.md",
      "source_sha256": "3b59d1181a0d1e755f09bcfffa4bb460ce48a7f6865a8eb6d8b523fe51dff7c0",
      "status": "author_self_checked"
    },
    {
      "id": "dendrites-and-cut-points",
      "title": "Dendrites and cut points",
      "reader": "dendrites-and-cut-points.html",
      "source": "src/dendrites-and-cut-points.md",
      "source_sha256": "dd00c3fddabd957cc26e6b730858989af2f741eb853eedef62b38d938d8e879e",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Point-Set Topology (C90): compactness, connectedness, complete metric spaces",
    "covering dimension: Definition 2.1 and Theorems 3.2, 4.2 of the lesson Covering dimension and finite trivializing covers (course Index theory of elliptic operators)",
    "the course proves every other fact it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}