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  1. Topology

    Linked via "connected"

    Connectedness
    A topological space $X$ is connected if it cannot be written as the union of two non-empty, disjoint open sets. If a space is not connected, its components are the maximal connected subspaces. A related, more nuanced concept is path-connectedness, where any two points can be joined by a continuous path lying entirely within the space. Whi…

  2. Topology

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    The Genus
    The genus of a connected, compact surface without boundary is a primary topological invariant. It quantifies the number of "handles" or "through-holes" in the object. For example, the sphere has genus 0, the torus (donut shape) has genus 1, and a double torus has genus 2.
    The relationship between [g…

  3. Torus

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    Topological Properties
    The torus is characterized by its topological invariants. It is a compact, connected, and orientable two-manifold.
    Genus and Euler Characteristic