"accumulation point" meaning in English

{
  "forms": [
    {
      "form": "accumulation points",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "accumulation point (plural accumulation points)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematical analysis",
          "orig": "en:Mathematical analysis",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Systems theory",
          "orig": "en:Systems theory",
          "parents": [
            "Sciences",
            "Systems",
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            "Interdisciplinary fields",
            "Society",
            "Fundamental"
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Topology",
          "orig": "en:Topology",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1975, Bert Mendelson, Introduction to Topology, 3rd edition, New York: Dover Publications, Inc., published 1990, →OCLC, §5.3, page 173",
          "text": "LEMMA 5.2 Let X be a Hausdorff space and A a subset of X. A point a#x5C;inX is an accumulation point of A if and only if a is a limit point of A.",
          "type": "quotation"
        },
        {
          "text": "2008, Brian S. Thomson, Andrew M. Bruckner, Judith B. Bruckner, Elementary Real Analysis, Volume 1, Thomson-Bruckner (ClassicalRealAnalysis.com), 2nd Edition, page 153,\nDefinition 4.9 (Closed): The set E is said to be closed provided that every accumulation point of E belongs to the set E.\nThus a set E is not closed if there is some accumulation point of E that does not belong to E. In particular, a set with no accumulation points would have to be closed since there is no point that needs to be checked."
        },
        {
          "text": "{{quote-book|en|year=2016|author=Jonathan M. Kane|title=Writing Proofs in Analysis|pageurl=https://books.google.com.au/books?id=Hm1BDAAAQBAJ&pg=PA74&dq=%22accumulation+point%22%7C%22accumulation+points%22&hl=en&sa=X&ved=2ahUKEwjXmIa0qoPrAhVWWysKHSMKDhAQ6AEwAnoECAYQAg#v=onepage&q=%22accumulation%20point%22%7C%22accumulation%20points%22&f=false|page=74|publisher=Springer"
        }
      ],
      "glosses": [
        "Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S."
      ],
      "id": "en-accumulation_point-en-noun-vTXDinDp",
      "links": [
        [
          "topology",
          "topology"
        ],
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          "topological space",
          "topological space"
        ],
        [
          "neighborhood",
          "neighborhood"
        ]
      ],
      "qualifier": "\"of\" a subset of a topological space",
      "raw_glosses": [
        "(topology, \"of\" a subset of a topological space) Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S."
      ],
      "synonyms": [
        {
          "word": "cluster point"
        },
        {
          "word": "limit point"
        }
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "Encyclopedia of Mathematics",
        "MathWorld"
      ]
    }
  ],
  "word": "accumulation point"
}

{
  "forms": [
    {
      "form": "accumulation points",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "accumulation point (plural accumulation points)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Mathematical analysis",
        "en:Systems theory",
        "en:Topology"
      ],
      "examples": [
        {
          "ref": "1975, Bert Mendelson, Introduction to Topology, 3rd edition, New York: Dover Publications, Inc., published 1990, →OCLC, §5.3, page 173",
          "text": "LEMMA 5.2 Let X be a Hausdorff space and A a subset of X. A point a#x5C;inX is an accumulation point of A if and only if a is a limit point of A.",
          "type": "quotation"
        },
        {
          "text": "2008, Brian S. Thomson, Andrew M. Bruckner, Judith B. Bruckner, Elementary Real Analysis, Volume 1, Thomson-Bruckner (ClassicalRealAnalysis.com), 2nd Edition, page 153,\nDefinition 4.9 (Closed): The set E is said to be closed provided that every accumulation point of E belongs to the set E.\nThus a set E is not closed if there is some accumulation point of E that does not belong to E. In particular, a set with no accumulation points would have to be closed since there is no point that needs to be checked."
        },
        {
          "text": "{{quote-book|en|year=2016|author=Jonathan M. Kane|title=Writing Proofs in Analysis|pageurl=https://books.google.com.au/books?id=Hm1BDAAAQBAJ&pg=PA74&dq=%22accumulation+point%22%7C%22accumulation+points%22&hl=en&sa=X&ved=2ahUKEwjXmIa0qoPrAhVWWysKHSMKDhAQ6AEwAnoECAYQAg#v=onepage&q=%22accumulation%20point%22%7C%22accumulation%20points%22&f=false|page=74|publisher=Springer"
        }
      ],
      "glosses": [
        "Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S."
      ],
      "links": [
        [
          "topology",
          "topology"
        ],
        [
          "topological space",
          "topological space"
        ],
        [
          "neighborhood",
          "neighborhood"
        ]
      ],
      "qualifier": "\"of\" a subset of a topological space",
      "raw_glosses": [
        "(topology, \"of\" a subset of a topological space) Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S."
      ],
      "synonyms": [
        {
          "word": "cluster point"
        },
        {
          "word": "limit point"
        }
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "Encyclopedia of Mathematics",
        "MathWorld"
      ]
    }
  ],
  "word": "accumulation point"
}