Chapter: Basic Topological Concepts

We start with some basic concepts of topology. Please note that the term topology has double meaning. The first meaning of this term is mathematical discipline "topology" itself. But "topology" also means - in its second meaning - one of the key concepts in this discipline. We will use the term sometimes in the first and sometimes in the second meaning, and it will be clear from the context, which meaning is meant.

Explanations: 1

  1. Definition: Topological Space, Topology
  2. Proposition: Clopen Sets and Boundaries
  3. Definition: Discrete and Indiscrete Topology
  4. Definition: Ordering of Topologies
  5. Definition: Open, Closed, Clopen
  6. Proposition: Alternative Characterization of Topological Spaces
  7. Definition: Neighborhood
  8. Proposition: Properties of the Set of All Neighborhoods of a Point
  9. Proposition: A Necessary Condition of a Neighborhood to be Open
  10. Definition: Boundary Points, Closures, Interiors, and Exteriors
  11. Proposition: How the Boundary Changes the Property of a Set of Being Open
  12. Definition: Regular Open, Regular Closed

Chapters: 1

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