{
  "schema": "bounded-prerequisite-component/v1",
  "title": "Positive Riesz representation and its topology and measure prerequisites",
  "component_id": "AN01-U008-POSITIVE-RIESZ",
  "mathematical_provider_id": "OA-FND-HM-01",
  "prepared_at": "2026-10-03T22:04:48.612633+00:00",
  "selection_and_link_repair": "GPT-6.1 Sol (OpenAI), Ultra; AN-01 course project",
  "license": "CC0-1.0 for selected original programme exposition",
  "sources": [
    {
      "source_checkout_relative": "docs/courses/harmonic-analysis-on-locally-compact-groups/src/haar-measure-on-locally-compact-groups.md",
      "source_url": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/harmonic-analysis-on-locally-compact-groups/src/haar-measure-on-locally-compact-groups.md",
      "full_source_sha256": "c3d938e2f10056cc392605c1b811fbc9399eca53104287036a43605c0bd17ad0",
      "full_source_bytes": 143387,
      "selected_ranges": [
        {
          "first_line": 1,
          "last_line": 96,
          "utf8_lf_final_newline_sha256": "bc426c8c0a4e47a473ab8a514eaeeed36f367b5226f20f5ae1d5036cd3857b20"
        },
        {
          "first_line": 797,
          "last_line": 804,
          "utf8_lf_final_newline_sha256": "b0ce7c4f92a29383d0f2ee7754ed857705d8487e6605873ea434f68b3cbe240a"
        }
      ],
      "live_public_source_read_check": {
        "url": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/harmonic-analysis-on-locally-compact-groups/src/haar-measure-on-locally-compact-groups.md",
        "status": 200,
        "bytes": 143387,
        "sha256": "c3d938e2f10056cc392605c1b811fbc9399eca53104287036a43605c0bd17ad0",
        "observed_at": "2026-10-03T22:03:52.9828619+00:00"
      },
      "component_source": "prerequisites/riesz/positive-riesz.md",
      "source_header_verbatim": "*Originally written by Claude Opus 5.5 (Anthropic), September 2026, with a separate historical AI spot-check of that text. Repaired and self-checked by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. The current edition has no separately recorded independent AI or human review. Original programme content is dedicated under CC0.*",
      "required_original_range": [
        21,
        96
      ],
      "selected_source_blocks": [
        {
          "original_lines": [
            1,
            3
          ],
          "component_lines": [
            1,
            3
          ]
        },
        {
          "original_lines": [
            4,
            96
          ],
          "component_lines": [
            7,
            99
          ]
        }
      ],
      "link_destination_changes": [
        {
          "from_markdown_href": "measure-and-hilbert-space-tools.md",
          "to_markdown_href": "measure-tools.md",
          "occurrences": 2
        },
        {
          "from_markdown_href": "measure-and-hilbert-space-tools.md#1-from-an-outer-measure-to-a-measure",
          "to_markdown_href": "measure-tools.md#1-from-an-outer-measure-to-a-measure",
          "occurrences": 1
        },
        {
          "from_markdown_href": "course:function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity",
          "to_markdown_href": "topology-cutoffs.md",
          "occurrences": 1
        },
        {
          "from_markdown_href": "course:function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity#oa-fnd-sw-15",
          "to_markdown_href": "topology-cutoffs.md#oa-fnd-sw-15",
          "occurrences": 1
        },
        {
          "from_markdown_href": "course:function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity#oa-fnd-sw-16",
          "to_markdown_href": "topology-cutoffs.md#oa-fnd-sw-16",
          "occurrences": 1
        },
        {
          "from_markdown_href": "course:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators",
          "to_markdown_href": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
          "occurrences": 1
        },
        {
          "from_markdown_href": "course:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
          "to_markdown_href": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html",
          "occurrences": 1
        }
      ],
      "only_source_text_changes": [
        "Markdown link destinations listed above",
        "line endings normalized to LF with final newline"
      ],
      "bibliography_range": [
        797,
        804
      ],
      "license": "CC0-1.0"
    },
    {
      "source_checkout_relative": "docs/courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "source_url": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "full_source_sha256": "ee22f2f44dd2c089b2bfb10d25fd5b36ebd9d7c245ea3206a1abb71e393e2e12",
      "full_source_bytes": 29235,
      "selected_ranges": [
        {
          "first_line": 1,
          "last_line": 72,
          "utf8_lf_final_newline_sha256": "278cc9e824a4d664bda7314d8cd97a39ae442b0fa86f8f292ef3d1c1cec6f2b1"
        },
        {
          "first_line": 226,
          "last_line": 237,
          "utf8_lf_final_newline_sha256": "00e5208b7e1a524fe953898b8f537114364eeb5d5781d7e46ccd96b192e6be34"
        }
      ],
      "live_public_source_read_check": {
        "url": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
        "status": 200,
        "bytes": 29235,
        "sha256": "ee22f2f44dd2c089b2bfb10d25fd5b36ebd9d7c245ea3206a1abb71e393e2e12",
        "observed_at": "2026-10-03T22:03:53.1532071+00:00"
      },
      "component_source": "prerequisites/riesz/measure-tools.md",
      "source_header_verbatim": "*Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Self-checked by the writing AI, GPT-6.1 Sol. Independent AI review remains separate. Original exposition is public domain (CC0).*",
      "required_original_range": [
        1,
        72
      ],
      "selected_source_blocks": [
        {
          "original_lines": [
            1,
            3
          ],
          "component_lines": [
            1,
            3
          ]
        },
        {
          "original_lines": [
            4,
            72
          ],
          "component_lines": [
            7,
            75
          ]
        }
      ],
      "link_destination_changes": [
        {
          "from_markdown_href": "course:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators",
          "to_markdown_href": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
          "occurrences": 1
        }
      ],
      "only_source_text_changes": [
        "Markdown link destinations listed above",
        "line endings normalized to LF with final newline"
      ],
      "bibliography_range": [
        226,
        237
      ],
      "license": "CC0-1.0"
    },
    {
      "source_checkout_relative": "docs/courses/function-algebras-and-approximation/src/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md",
      "source_url": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/function-algebras-and-approximation/src/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md",
      "full_source_sha256": "c8cf4e39fcb24f9add4d693a1e53842f3c7ed04a6fb0bf98bf55b1719fb2da3b",
      "full_source_bytes": 124955,
      "selected_ranges": [
        {
          "first_line": 1,
          "last_line": 172,
          "utf8_lf_final_newline_sha256": "7a6b3e8d036bd542d39ec4be507d4054de48fbd3268a776552a12a49c1a0df2d"
        },
        {
          "first_line": 718,
          "last_line": 734,
          "utf8_lf_final_newline_sha256": "7dfb6be32923b936510c1d97a818bf7c6752c94304f9d420e413771b51bd4a60"
        }
      ],
      "live_public_source_read_check": {
        "url": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/function-algebras-and-approximation/src/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md",
        "status": 200,
        "bytes": 124955,
        "sha256": "c8cf4e39fcb24f9add4d693a1e53842f3c7ed04a6fb0bf98bf55b1719fb2da3b",
        "observed_at": "2026-10-03T22:03:53.1737991+00:00"
      },
      "component_source": "prerequisites/riesz/topology-cutoffs.md",
      "source_header_verbatim": "*Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session. The October revisions correct two points found by an AI reviewer in a separate OpenAI Codex session and add proofs of the background facts that the core courses leave as exercises; they are self-checked by the writing AI. Public domain (CC0).*",
      "required_original_range": [
        1,
        172
      ],
      "selected_source_blocks": [
        {
          "original_lines": [
            1,
            3
          ],
          "component_lines": [
            1,
            3
          ]
        },
        {
          "original_lines": [
            4,
            172
          ],
          "component_lines": [
            7,
            175
          ]
        }
      ],
      "link_destination_changes": [
        {
          "from_markdown_href": "#oa-fnd-sw-09",
          "to_markdown_href": "https://kokunoyumeto.github.io/open-math-courses/courses/function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.html#oa-fnd-sw-09",
          "occurrences": 2
        }
      ],
      "only_source_text_changes": [
        "Markdown link destinations listed above",
        "line endings normalized to LF with final newline"
      ],
      "bibliography_range": [
        718,
        734
      ],
      "license": "CC0-1.0"
    }
  ],
  "upstream_historical_credits": {
    "stone_authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session. The October revisions correct two points found by an AI reviewer in a separate OpenAI Codex session and add proofs of the background facts that the core courses leave as exercises; they are self-checked by the writing AI. Public domain (CC0).",
    "stone_source_use": {
      "background": "Exact Lebl and Levin results listed in the lesson; the remaining linearity, complex integration and exponential steps are proved locally.",
      "historical_and_comparative": "Stone, Weierstrass, Urysohn, Alexandroff, Bishop and Altomare; full references appear in the lesson.",
      "further_reading": "Blackadar and van Neerven; pi-Base for counterexamples."
    },
    "stone_packaging": {
      "by": "GPT-6 Astra (OpenAI), Ultra",
      "date": "2026-10-03",
      "description": "Complete October 2026 lesson with programme links to its prerequisites, course navigation, reproducible reader and offline download.",
      "independent_mathematical_review_added": false
    },
    "stone_licence": "CC0-1.0 for original lesson text. Linked human works and bundled software retain their own terms.",
    "haar_ai_authorship_record": {
      "haar_lesson": "Claude Opus 5.5 (Anthropic)",
      "quotient_lesson": "GPT-6 Astra (OpenAI), Ultra",
      "measure_tools_arithmetic_source_integration_and_self_check": "GPT-6.1 Sol (OpenAI), Ultra",
      "source_comparison_and_proof_expansions": "GPT-6.1 Sol (OpenAI), Codex, Ultra"
    },
    "haar_current_review_record": {
      "writing_instance_checked": true,
      "independent_ai_review": false,
      "human_review": false,
      "formal_verification": false
    },
    "source_headers": [
      "*Originally written by Claude Opus 5.5 (Anthropic), September 2026, with a separate historical AI spot-check of that text. Repaired and self-checked by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. The current edition has no separately recorded independent AI or human review. Original programme content is dedicated under CC0.*",
      "*Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Self-checked by the writing AI, GPT-6.1 Sol. Independent AI review remains separate. Original exposition is public domain (CC0).*",
      "*Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session. The October revisions correct two points found by an AI reviewer in a separate OpenAI Codex session and add proofs of the background facts that the core courses leave as exercises; they are self-checked by the writing AI. Public domain (CC0).*"
    ]
  },
  "cc0_license_source": {
    "source_checkout_relative": "docs/courses/function-algebras-and-approximation/LICENSE-CC0.txt",
    "sha256": "437ca31a127c9f1b191ff9a42bb563e5f7e414ae819006dca4635d9fb34aeb94",
    "copied_without_byte_changes": true
  },
  "required_chain": [
    "Positive Riesz: topology tools T1–T3 plus Carathéodory and nonnegative integration",
    "Topology: conventions and elementary compactness facts, one-point compactification, compact Hausdorff normality, local shrinking, Urysohn's lemma and compact cutoffs; all full proofs selected",
    "Measure: Carathéodory, elementary integral construction, continuity from below, monotone convergence, Fatou and dominated convergence; all full proofs selected"
  ],
  "excluded_unneeded_source": "Later Haar sections and the Fremlin-following neighbourhood-uniqueness supplement; later Hilbert, Lp-duality and Stone–Weierstrass proofs",
  "optional_external_context": [
    {
      "from": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "href": "https://github.com/jirilebl/ra/blob/e21ec524ca7d54f800c693b948020c188d21d01f/ch-metric.tex#L2217-L2406",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "href": "https://github.com/leanprover-community/mathlib4/blob/71a80585ee495fc24472fd0eaffc89d94e4fd8d6/Mathlib/Analysis/Normed/Module/HahnBanach.lean",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "href": "https://www-users.cse.umn.edu/~garrett/m/fun/notes_2012-13/01_metric_spaces.pdf",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "href": "https://www-users.cse.umn.edu/~garrett/m/fun/notes_2016-17/02_banach.pdf",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/complex-hahn-banach/HISTORY.md",
      "href": "https://kokunoyumeto.github.io/open-math-courses/courses/KT-CP/prerequisites/AN-03/banach-foundation-bridges.html#AN03-BFD-005",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/complex-hahn-banach/HISTORY.md",
      "href": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/KT-CP/prerequisites/src/AN-03/banach-foundation-bridges.md",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://fr.wikisource.org/wiki/Leçons_sur_l’intégration_et_la_recherche_des_fonctions_primitives_(première_édition)",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-C10",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-C20",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-D10",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://measure.axler.net/MIRA.pdf",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://orthogonalpublishing.com/lnfa/html/hilbertschmidt.html",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://web.pdx.edu/~erdman/",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://www.jirka.org/ra/",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/measure-tools.md",
      "href": "https://www1.essex.ac.uk/maths/people/fremlin/mt.htm",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://arxiv.org/abs/2102.07636v1",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://biblio.ugent.be/publication/8585474",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-C90",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-D10",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/Unique.html",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/harmonic-analysis-on-locally-compact-groups/src/haar-measure-on-locally-compact-groups.md",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://www1.essex.ac.uk/maths/people/fremlin/mt.htm",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/positive-riesz.md",
      "href": "https://www1.essex.ac.uk/maths/people/fremlin/mtcont.htm",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://discrete.openmathbooks.org/dmoi4/",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://kokunoyumeto.github.io/open-math-courses/courses/function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.html#oa-fnd-sw-09",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-C10",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-C20",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-C90",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/B10/reader/",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://raw.githubusercontent.com/KokunoYumeto/open-math-courses/main/docs/courses/function-algebras-and-approximation/src/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://topology.pi-base.org/",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://topology.pi-base.org/properties/P000023",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://www.jirka.org/ra/",
      "role": "external source, citation or context; no selected proof dependency"
    },
    {
      "from": "prerequisites/riesz/topology-cutoffs.md",
      "href": "https://www.jirka.org/ra/html/sec_stoneweier.html",
      "role": "external source, citation or context; no selected proof dependency"
    }
  ],
  "heading_anchors": {
    "prerequisites/riesz/positive-riesz.md": [
      {
        "line": 48,
        "heading": "## 2. Radon measures and the Riesz representation theorem",
        "ids": [
          "OA-FND-HM-01",
          "oa-fnd-hm-01"
        ]
      }
    ],
    "prerequisites/riesz/measure-tools.md": [
      {
        "line": 12,
        "heading": "## 1. From an outer measure to a measure",
        "ids": [
          "1-from-an-outer-measure-to-a-measure"
        ]
      },
      {
        "line": 39,
        "heading": "## 2. Integration and convergence without countability assumptions",
        "ids": [
          "2-integration-and-convergence-without-countability-assumptions"
        ]
      }
    ],
    "prerequisites/riesz/topology-cutoffs.md": [
      {
        "line": 39,
        "heading": "## 2. Spaces of bounded and continuous functions",
        "ids": [
          "OA-FND-SW-01",
          "oa-fnd-sw-01"
        ]
      },
      {
        "line": 65,
        "heading": "## 3. The one-point compactification",
        "ids": [
          "OA-FND-SW-02",
          "OA-FND-SW-03",
          "oa-fnd-sw-02",
          "oa-fnd-sw-03"
        ]
      },
      {
        "line": 116,
        "heading": "## 4. Compact and locally compact spaces",
        "ids": [
          "OA-FND-SW-15",
          "oa-fnd-sw-15"
        ]
      },
      {
        "line": 155,
        "heading": "## 5. Urysohn's lemma and metrizability",
        "ids": [
          "OA-FND-SW-16",
          "oa-fnd-sw-16"
        ]
      }
    ]
  },
  "markdown_to_html": {
    "prerequisites/complex-hahn-banach/AN03-P004-005.md": "prerequisites/complex-hahn-banach/AN03-P004-005.html",
    "prerequisites/complex-hahn-banach/HISTORY.md": "prerequisites/complex-hahn-banach/HISTORY.html",
    "prerequisites/complex-hahn-banach/RIGHTS.md": "prerequisites/complex-hahn-banach/RIGHTS.html",
    "prerequisites/complex-hahn-banach/TITLE_PAGE.md": "prerequisites/complex-hahn-banach/TITLE_PAGE.html",
    "prerequisites/riesz/measure-tools.md": "prerequisites/riesz/measure-tools.html",
    "prerequisites/riesz/positive-riesz.md": "prerequisites/riesz/positive-riesz.html",
    "prerequisites/riesz/topology-cutoffs.md": "prerequisites/riesz/topology-cutoffs.html"
  },
  "offline_link_mapping": [
    {
      "from_markdown": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "markdown_href": "AN03-P004-005.md",
      "generated_reader_href": "AN03-P004-005.md",
      "archive_target": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "fragment": null,
      "purpose": "transparent Markdown source; keep this .md href"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "markdown_href": "COPYING",
      "generated_reader_href": "COPYING",
      "archive_target": "prerequisites/complex-hahn-banach/COPYING",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "markdown_href": "HISTORY.md",
      "generated_reader_href": "HISTORY.html",
      "archive_target": "prerequisites/complex-hahn-banach/HISTORY.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "markdown_href": "RIGHTS.md",
      "generated_reader_href": "RIGHTS.html",
      "archive_target": "prerequisites/complex-hahn-banach/RIGHTS.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "markdown_href": "TITLE_PAGE.md",
      "generated_reader_href": "TITLE_PAGE.html",
      "archive_target": "prerequisites/complex-hahn-banach/TITLE_PAGE.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/HISTORY.md",
      "markdown_href": "AN03-P004-005.md",
      "generated_reader_href": "AN03-P004-005.md",
      "archive_target": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "fragment": null,
      "purpose": "transparent Markdown source; keep this .md href"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/HISTORY.md",
      "markdown_href": "component-provenance.json",
      "generated_reader_href": "component-provenance.json",
      "archive_target": "prerequisites/complex-hahn-banach/component-provenance.json",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/RIGHTS.md",
      "markdown_href": "AN03-P004-005.md",
      "generated_reader_href": "AN03-P004-005.md",
      "archive_target": "prerequisites/complex-hahn-banach/AN03-P004-005.md",
      "fragment": null,
      "purpose": "transparent Markdown source; keep this .md href"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/RIGHTS.md",
      "markdown_href": "COPYING",
      "generated_reader_href": "COPYING",
      "archive_target": "prerequisites/complex-hahn-banach/COPYING",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/RIGHTS.md",
      "markdown_href": "HISTORY.md",
      "generated_reader_href": "HISTORY.html",
      "archive_target": "prerequisites/complex-hahn-banach/HISTORY.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/RIGHTS.md",
      "markdown_href": "TITLE_PAGE.md",
      "generated_reader_href": "TITLE_PAGE.html",
      "archive_target": "prerequisites/complex-hahn-banach/TITLE_PAGE.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/RIGHTS.md",
      "markdown_href": "component-provenance.json",
      "generated_reader_href": "component-provenance.json",
      "archive_target": "prerequisites/complex-hahn-banach/component-provenance.json",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/TITLE_PAGE.md",
      "markdown_href": "COPYING",
      "generated_reader_href": "COPYING",
      "archive_target": "prerequisites/complex-hahn-banach/COPYING",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/TITLE_PAGE.md",
      "markdown_href": "HISTORY.md",
      "generated_reader_href": "HISTORY.html",
      "archive_target": "prerequisites/complex-hahn-banach/HISTORY.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/complex-hahn-banach/TITLE_PAGE.md",
      "markdown_href": "RIGHTS.md",
      "generated_reader_href": "RIGHTS.html",
      "archive_target": "prerequisites/complex-hahn-banach/RIGHTS.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/measure-tools.md",
      "markdown_href": "LICENSE-CC0.txt",
      "generated_reader_href": "LICENSE-CC0.txt",
      "archive_target": "prerequisites/riesz/LICENSE-CC0.txt",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/measure-tools.md",
      "markdown_href": "component-provenance.json",
      "generated_reader_href": "component-provenance.json",
      "archive_target": "prerequisites/riesz/component-provenance.json",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/positive-riesz.md",
      "markdown_href": "LICENSE-CC0.txt",
      "generated_reader_href": "LICENSE-CC0.txt",
      "archive_target": "prerequisites/riesz/LICENSE-CC0.txt",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/positive-riesz.md",
      "markdown_href": "component-provenance.json",
      "generated_reader_href": "component-provenance.json",
      "archive_target": "prerequisites/riesz/component-provenance.json",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/positive-riesz.md",
      "markdown_href": "measure-tools.md",
      "generated_reader_href": "measure-tools.html",
      "archive_target": "prerequisites/riesz/measure-tools.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/positive-riesz.md",
      "markdown_href": "measure-tools.md#1-from-an-outer-measure-to-a-measure",
      "generated_reader_href": "measure-tools.html#1-from-an-outer-measure-to-a-measure",
      "archive_target": "prerequisites/riesz/measure-tools.html",
      "fragment": "1-from-an-outer-measure-to-a-measure",
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/positive-riesz.md",
      "markdown_href": "topology-cutoffs.md",
      "generated_reader_href": "topology-cutoffs.html",
      "archive_target": "prerequisites/riesz/topology-cutoffs.html",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/positive-riesz.md",
      "markdown_href": "topology-cutoffs.md#oa-fnd-sw-15",
      "generated_reader_href": "topology-cutoffs.html#oa-fnd-sw-15",
      "archive_target": "prerequisites/riesz/topology-cutoffs.html",
      "fragment": "oa-fnd-sw-15",
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/positive-riesz.md",
      "markdown_href": "topology-cutoffs.md#oa-fnd-sw-16",
      "generated_reader_href": "topology-cutoffs.html#oa-fnd-sw-16",
      "archive_target": "prerequisites/riesz/topology-cutoffs.html",
      "fragment": "oa-fnd-sw-16",
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/topology-cutoffs.md",
      "markdown_href": "#oa-fnd-sw-02",
      "generated_reader_href": "#oa-fnd-sw-02",
      "archive_target": "prerequisites/riesz/topology-cutoffs.html",
      "fragment": "oa-fnd-sw-02",
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/topology-cutoffs.md",
      "markdown_href": "#oa-fnd-sw-15",
      "generated_reader_href": "#oa-fnd-sw-15",
      "archive_target": "prerequisites/riesz/topology-cutoffs.html",
      "fragment": "oa-fnd-sw-15",
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/topology-cutoffs.md",
      "markdown_href": "LICENSE-CC0.txt",
      "generated_reader_href": "LICENSE-CC0.txt",
      "archive_target": "prerequisites/riesz/LICENSE-CC0.txt",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    },
    {
      "from_markdown": "prerequisites/riesz/topology-cutoffs.md",
      "markdown_href": "component-provenance.json",
      "generated_reader_href": "component-provenance.json",
      "archive_target": "prerequisites/riesz/component-provenance.json",
      "fragment": null,
      "purpose": "reader or accompanying notice/provenance"
    }
  ],
  "independent_mathematical_certification": false
}