Chapter: Continuity

Functions on topological spaces help to study their properties and to construct new spaces from previously existing ones. The most important type of functions in topology are continuous functions. Not surprisingly, they can be defined for general topological spaces, without making use of a whatsoever "distance" concept. This chapter demonstrates how. Also, the important concept of homeomorphism is introduced.

Examples: 1

  1. Definition: Continuous Function
  2. Proposition: Equivalent Notions of Continuous Functions
  3. Proposition: Continuity of Compositions of Functions
  4. Definition: Open and Closed Functions
  5. Proposition: Bijective Open Functions
  6. Definition: Homeomorphism, Homeomorphic Spaces
  7. Proposition: Equivalent Notions of Homeomorphisms
  8. Definition: Topological, Continuous, Open, and Closed Invariants

Parts: 1

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  1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
  2. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition