analysis situs

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Definition
  1. Noun:
    • Topology: The branch of pure mathematics concerned with the properties of geometric figures that remain unchanged even when the figure is stretched, bent, or otherwise deformed, as long as it is not torn or glued. It deals with concepts like continuity, connectedness, and boundaries.
Usage Examples
  • Noun:
    • The mathematician specialized in analysis situs, studying the fundamental properties of surfaces.
    • Problems in analysis situs often involve distinguishing between shapes like a coffee cup and a doughnut, which are considered equivalent in this field.
    • The course introduced the basic principles of analysis situs, such as homeomorphism and homotopy.
Advanced Usage
  • The term "analysis situs" is considered an older, historical name for the field now universally known as topology. In modern academic writing, "topology" is the standard term.
    • Henri Poincaré is often called the father of modern analysis situs.
    • The paper traced the evolution of ideas from analysis situs to algebraic topology.
Variants and Related Words
  • Topology (n): The modern term for analysis situs. It has subfields like:
    • General topology (or point-set topology): Studies fundamental set-theoretic definitions like open sets, continuity, and convergence.
    • Algebraic topology: Uses tools from abstract algebra to classify topological spaces.
    • Geometric topology: Studies manifolds and their embeddings.
Synonyms
  • Topology: The direct and modern synonym.
Related Concepts (Not Phrasal Verbs or Idioms)
  • Homeomorphism: A one-to-one correspondence between two spaces that is continuous in both directions, which is the core concept defining equivalence in analysis situs/topology.
  • Invariant: A property (like the number of holes) that remains unchanged under homeomorphisms.
  • Manifold: A topological space that locally resembles Euclidean space.
Noun
  1. the branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a one-to-one correspondence that is continuous in both directions

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