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  2. Algebraic topology - Wikipedia

    en.wikipedia.org/wiki/Algebraic_topology

    Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological ...

  3. Topology - Wikipedia

    en.wikipedia.org/wiki/Topology

    A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...

  4. Homology (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Homology_(mathematics)

    In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages.The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups.

  5. General topology - Wikipedia

    en.wikipedia.org/wiki/General_topology

    In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology , geometric topology , and algebraic topology .

  6. Algebraic geometry and analytic geometry - Wikipedia

    en.wikipedia.org/wiki/Algebraic_geometry_and...

    The topology on X an is called the "complex topology" (and is very different from the subspace topology). Suppose φ: X → Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map φ an: X an → Y an such that λ Y ∘ φ an = φ ∘ λ X.

  7. Topological group - Wikipedia

    en.wikipedia.org/wiki/Topological_group

    The real numbers form a topological group under addition. In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.

  8. Topological space - Wikipedia

    en.wikipedia.org/wiki/Topological_space

    In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...

  9. Homotopy theory - Wikipedia

    en.wikipedia.org/wiki/Homotopy_theory

    In homotopy theory and algebraic topology, the word "space" denotes a topological space.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.