University · Mathematics · Topology

Connectedness and Compactness

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A rigorous treatment of two of the most important qualitative properties in topology: connectedness (a space that cannot be split into two disjoint nonempty open sets) and compactness (the open-cover generalization of 'closed and bounded'), including path-connectedness, the topologist's sine curve, the Heine–Borel theorem, and why compactness is strictly stronger than closed-and-bounded outside of R^n.

Inhaltsübersicht

  • Separations and the Definition of Connectedness
  • Path-Connectedness and the Topologist's Sine Curve
  • Compactness: Open Covers and Basic Properties
  • The Heine–Borel Theorem and the Limits of Closed-and-Bounded
Diagram of a topological space split into two disjoint open sets forming a separation
Pixabay – Pixabay License

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