{
  "id": "pretrees",
  "title": "Pretrees",
  "description": "Sets with a betweenness relation that behaves as in a tree: the axioms, intervals, the theorem that every finite pretree is a set of vertices of a finite tree, directions on linear sets, uniqueness of medians, and dense, complete and median pretrees, with an example of a complete dense pretree that is not median, separable pretrees, whose intervals are ordered like the real interval [0,1], a theorem that recognizes dendrites by a pretree structure, flows, directed arcs with their suprema, the completion of a pretree by its flows, the three kinds of flows, the adjacent points of the completion, discrete pretrees as sets of vertices of trees, quotients of pretrees, the dense quotient, the pretree of cut points of a connected space, the map from such a space to the completion of its set of cut points, and the dendrite quotient of a continuum.",
  "units": [
    {
      "id": "betweenness-and-pretrees",
      "title": "Betweenness and pretrees",
      "reader": "betweenness-and-pretrees.html",
      "source": "src/betweenness-and-pretrees.md",
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      "status": "author_self_checked"
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    {
      "id": "dense-complete-and-median-pretrees",
      "title": "Dense, complete and median pretrees",
      "reader": "dense-complete-and-median-pretrees.html",
      "source": "src/dense-complete-and-median-pretrees.md",
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      "status": "author_self_checked"
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    {
      "id": "separable-pretrees-and-the-order-of-intervals",
      "title": "Separable pretrees and the order of intervals",
      "reader": "separable-pretrees-and-the-order-of-intervals.html",
      "source": "src/separable-pretrees-and-the-order-of-intervals.md",
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    },
    {
      "id": "dendrites-from-pretrees",
      "title": "Dendrites from pretrees",
      "reader": "dendrites-from-pretrees.html",
      "source": "src/dendrites-from-pretrees.md",
      "source_sha256": "52818dfb2706828a1b54dd40022f5e3c4ff84f1b73ae462b88f58ca5a592ee29",
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    },
    {
      "id": "flows-on-a-pretree",
      "title": "Flows on a pretree",
      "reader": "flows-on-a-pretree.html",
      "source": "src/flows-on-a-pretree.md",
      "source_sha256": "6a9514d9094bd464793ab20df1cc7e575f437f33bd024986f2b525abd2c00b2a",
      "status": "author_self_checked"
    },
    {
      "id": "directed-arcs-and-suprema",
      "title": "Directed arcs and suprema",
      "reader": "directed-arcs-and-suprema.html",
      "source": "src/directed-arcs-and-suprema.md",
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    {
      "id": "the-completion-of-a-pretree",
      "title": "The completion of a pretree",
      "reader": "the-completion-of-a-pretree.html",
      "source": "src/the-completion-of-a-pretree.md",
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    },
    {
      "id": "stars-and-the-three-kinds-of-flows",
      "title": "Stars and the three kinds of flows",
      "reader": "stars-and-the-three-kinds-of-flows.html",
      "source": "src/stars-and-the-three-kinds-of-flows.md",
      "source_sha256": "7d8e846eb77238a64428409c8ace0b8a54fbbb0e6dd5eb100f0a82f43dbf0751",
      "status": "author_self_checked"
    },
    {
      "id": "adjacent-points-of-the-completion",
      "title": "Adjacent points of the completion",
      "reader": "adjacent-points-of-the-completion.html",
      "source": "src/adjacent-points-of-the-completion.md",
      "source_sha256": "ba96758a0c693f301887b612e3dc643b4b9898770b1507991b29147740bf1b4f",
      "status": "author_self_checked"
    },
    {
      "id": "discrete-pretrees-and-trees",
      "title": "Discrete pretrees and trees",
      "reader": "discrete-pretrees-and-trees.html",
      "source": "src/discrete-pretrees-and-trees.md",
      "source_sha256": "05845a6ee786e2eae26f65c0321809f7f080c9b865fd57161b4023f7f33d40e5",
      "status": "author_self_checked"
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    {
      "id": "quotients-of-pretrees",
      "title": "Quotients of pretrees",
      "reader": "quotients-of-pretrees.html",
      "source": "src/quotients-of-pretrees.md",
      "source_sha256": "2f60044b4fe33a4dd275686937614b234aeb3f40207e0b86e449c143ae3ba6cb",
      "status": "author_self_checked"
    },
    {
      "id": "the-dense-quotient-of-a-pretree",
      "title": "The dense quotient of a pretree",
      "reader": "the-dense-quotient-of-a-pretree.html",
      "source": "src/the-dense-quotient-of-a-pretree.md",
      "source_sha256": "27a2ae0d854cf7ba4570d8f47b015b2313e2e847a88911a3fde7e7bff81c3c9e",
      "status": "author_self_checked"
    },
    {
      "id": "cut-points-of-a-connected-space",
      "title": "Cut points of a connected space",
      "reader": "cut-points-of-a-connected-space.html",
      "source": "src/cut-points-of-a-connected-space.md",
      "source_sha256": "109ed7dc2528258ff6708d980cc6b4c15e67a6ac39d657cc5cbcb912fab3080f",
      "status": "author_self_checked"
    },
    {
      "id": "mapping-a-pretree-to-the-completion-of-a-subset",
      "title": "Mapping a pretree to the completion of a subset",
      "reader": "mapping-a-pretree-to-the-completion-of-a-subset.html",
      "source": "src/mapping-a-pretree-to-the-completion-of-a-subset.md",
      "source_sha256": "d8a31c9795259358e69194e402bb5c2a1c6976999e6fa8a325ffdadcccc011c7",
      "status": "author_self_checked"
    },
    {
      "id": "the-dendrite-quotient-of-a-continuum",
      "title": "The dendrite quotient of a continuum",
      "reader": "the-dendrite-quotient-of-a-continuum.html",
      "source": "src/the-dendrite-quotient-of-a-continuum.md",
      "source_sha256": "6b9fb126f8b9215e8e6bfc333b9910387e4a0729b73e9ae68322938c261446c5",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "graphs, paths, cycles and connected graphs (definitions only)",
    "for the fourth lesson: the course Continua, arcs and cut points (arcs, dendrites) and the Urysohn metrization theorem (core course Point-Set Topology)",
    "for the twelfth lesson: ordinal numbers, transfinite recursion and Hartogs' lemma (core course Mathematical Logic, Set Theory, and Computability)",
    "for the thirteenth lesson: connected, compact and Hausdorff spaces (core course Point-Set Topology)",
    "the course proves every other fact it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}