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Topological Relationships Between Point Sets: Disjointness, Coverage, and Overlap

This document explores the relationships between point sets within a topological framework. It delves into the definitions and implications of various operations, such as disjointness, intersection, coverage, and containment. Our analysis considers the significant relations including when two sets are disjoint, meet each other, are equal, or how they overlap. The goal is to provide a clear understanding of these topological concepts, supporting students and researchers in geometry and topology. The focus is on effective mathematical representation of these relations.

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Topological Relationships Between Point Sets: Disjointness, Coverage, and Overlap

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  1. Punktmengentopologie • Beziehungen zwischen Punktmengen

  2. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y X Y

  3. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y not     X meets Y X Y

  4. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y not     X meets Y not  not    X equals Y X Y

  5. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y not     X meets Y not  not    X equals Y  not  not   X inside Y X Y

  6. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y not     X meets Y not  not    X equals Y  not  not   X inside Y  not   not  Y inside X X Y

  7. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y not     X meets Y not  not    X equals Y  not  not   X inside Y  not   not  Y inside X not  not   not  X covers Y X Y

  8. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y not     X meets Y not  not    X equals Y  not  not   X inside Y  not   not  Y inside X not  not   not  X covers Y not  not  not   Y covers X X Y

  9. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjoint Y not     X meets Y not  not    X equals Y  not  not   X inside Y  not   not  Y inside X not  not   not  X covers Y not  not  not   Y covers X not  not  not  not  X overlaps Y X Y

  10. X  Y X°  Y° X  Y° X°  Y Operation Topologische Relationen     X disjunkt Y not     X trifft Y not  not    X gleicht Y  not  not   X innerhalb Y  not   not  Y innerhalb X not  not   not  X überdeckt Y not  not  not   Y überdeckt X not  not  not  not  X überlappt Y

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