Metamath Proof Explorer

Table of Contents - 12. BASIC TOPOLOGY

  1. Topology
    1. Topological spaces
    2. Topological bases
    3. Examples of topologies
    4. Closure and interior
    5. Neighborhoods
    6. Limit points and perfect sets
    7. Subspace topologies
    8. Order topology
    9. Limits and continuity in topological spaces
    10. Separated spaces: T0, T1, T2 (Hausdorff) ...
    11. Compactness
    12. Bolzano-Weierstrass theorem
    13. Connectedness
    14. First- and second-countability
    15. Local topological properties
    16. Refinements
    17. Compactly generated spaces
    18. Product topologies
    19. Continuous function-builders
    20. Quotient maps and quotient topology
    21. Homeomorphisms
  2. Filters and filter bases
    1. Filter bases
    2. Filters
    3. Ultrafilters
    4. Filter limits
    5. Extension by continuity
    6. Topological groups
    7. Infinite group sum on topological groups
    8. Topological rings, fields, vector spaces
  3. Uniform Structures and Spaces
    1. Uniform structures
    2. The topology induced by an uniform structure
    3. Uniform Spaces
    4. Uniform continuity
    5. Cauchy filters in uniform spaces
    6. Complete uniform spaces
  4. Metric spaces
    1. Pseudometric spaces
    2. Basic metric space properties
    3. Metric space balls
    4. Open sets of a metric space
    5. Continuity in metric spaces
    6. The uniform structure generated by a metric
    7. Examples of metric spaces
    8. Normed algebraic structures
    9. Normed space homomorphisms (bounded linear operators)
    10. Topology on the reals
    11. Topological definitions using the reals
    12. Path homotopy
    13. The fundamental group
  5. Metric subcomplex vector spaces
    1. Subcomplex modules
    2. Subcomplex vector spaces
    3. Normed subcomplex vector spaces
    4. Subcomplex pre-Hilbert spaces
    5. Neighborhoods and closure
    6. Convergence and completeness
    7. Baire's Category Theorem
    8. Banach spaces and subcomplex Hilbert spaces
    9. Euclidean spaces
    10. Minimizing Vector Theorem
    11. Projection Theorem