Math 2.0

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General topology

519 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 208 of its 212 ideas have a page so far.

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Map of the subject

Basic notions: open sets, neighbourhoods, bases 23 of 23

  1. Topological space definition
  2. Open set definition
  3. Closed set definition
  4. Clopen set definition
  5. Neighbourhood definition
  6. Basis of a topology definition
  7. Subbasis definition
  8. Comparison of topologies concept
  9. Interior definition
  10. Closure definition
  11. Boundary definition
  12. Kuratowski closure axioms definition
  13. Kuratowski's closure-complement problem problem
  14. Dense set definition
  15. Nowhere dense set definition
  16. Regular open set definition
  17. Limit point definition
  18. Isolated point definition
  19. Derived set definition
  20. Perfect set definition
  21. G-delta set definition
  22. F-sigma set definition
  23. Specialization preorder definition

Standard examples of topologies 10 of 10

  1. Discrete topology example
  2. Indiscrete topology example
  3. Cofinite topology example
  4. Sierpiński space example
  5. Particular point topology example
  6. Excluded point topology example
  7. Alexandrov topology definition
  8. Zariski topology example
  9. Order topology definition
  10. Scott topology definition

Filters, nets and convergence 6 of 6

  1. Filter definition
  2. Convergence of filters definition
  3. Ultrafilter definition
  4. Net definition
  5. Limit point definition
  6. Germ definition

Continuous maps 17 of 19

  1. Continuous map definition
  2. Continuity at a point definition
  3. Sequential continuity definition
  4. Homeomorphism definition
  5. Open map definition
  6. Closed map definition
  7. Topological embedding definition
  8. Quotient map definition
  9. Local homeomorphism definition
  10. Proper map definition
  11. Perfect map definition
  12. Semicontinuity definition
  13. Hemicontinuity definition
  14. Michael selection theorem statement
  15. intermediate value theorem statement
  16. extreme value theorem statement
  17. Left inverse definition
  18. Absolute neighborhood retract definition
  19. Dugundji extension theorem statement

Constructions of spaces 15 of 16

  1. Subspace topology definition
  2. Product topology definition
  3. Box topology definition
  4. Disjoint union topology definition
  5. quotient space definition
  6. Initial topology definition
  7. Final topology definition
  8. Adjunction space definition
  9. Cone definition
  10. Suspension definition
  11. Inverse limit definition
  12. Compact-open topology definition
  13. Topology of pointwise convergence definition
  14. Topology of uniform convergence definition
  15. Vietoris topology definition
  16. Hausdorff distance definition

Separation axioms 14 of 14

  1. Separation axioms concept
  2. T0 space definition
  3. Kolmogorov quotient definition
  4. T1 space definition
  5. Hausdorff space definition
  6. Urysohn space definition
  7. Regular space definition
  8. Tychonoff space definition
  9. Normal space definition
  10. Completely normal space definition
  11. Perfectly normal space definition
  12. Sober space definition
  13. Urysohn's lemma statement
  14. Tietze extension theorem statement

Countability axioms 6 of 6

  1. First-countable space definition
  2. Second-countable space definition
  3. Separable space definition
  4. Sequential space definition
  5. Fréchet–Urysohn space definition
  6. Cardinal function definition

Compactness 25 of 26

  1. Open cover definition
  2. Compact space definition
  3. finite intersection property definition
  4. Continuous image of a compact set is compact statement
  5. Compact subsets of a Hausdorff space are closed statement
  6. Compactness via ultrafilters statement
  7. Heine–Borel theorem statement
  8. Tychonoff's theorem statement
  9. Alexander subbase theorem statement
  10. Cantor's intersection theorem statement
  11. Lebesgue number lemma statement
  12. Tube lemma statement
  13. Locally compact space definition
  14. Sigma-compact space definition
  15. Countably compact space definition
  16. Sequential compactness definition
  17. Compactness and sequential compactness agree in metric spaces statement
  18. Limit point compact space definition
  19. Pseudocompact space definition
  20. Lindelöf space definition
  21. Locally finite family definition
  22. Paracompact space definition
  23. Stone's theorem on paracompactness of metric spaces statement
  24. Dieudonné's theorem on paracompact Hausdorff spaces statement
  25. Metacompact space definition
  26. Compactly generated space definition

Compactifications 7 of 7

  1. Compactification definition
  2. One-point compactification definition
  3. Stone–Čech compactification definition
  4. Stone space definition
  5. Stone's representation theorem for Boolean algebras statement
  6. Extremally disconnected space definition
  7. Gleason's theorem on projective compact spaces statement

Baire category 4 of 4

  1. Baire space definition
  2. Baire category theorem statement
  3. Meagre set definition
  4. Residual set definition

Connectedness 13 of 13

  1. Connected space definition
  2. Path-connected space definition
  3. Connected subsets of the real line are intervals statement
  4. Connected component definition
  5. Locally connected space definition
  6. Locally path-connected space definition
  7. Totally disconnected space definition
  8. Zero-dimensional space definition
  9. Hyperconnected space definition
  10. Noetherian topological space definition
  11. Spectral space definition
  12. Topologist's sine curve counterexample
  13. Comb space counterexample

Metric spaces 18 of 18

  1. Metric space definition
  2. Pseudometric space definition
  3. Ultrametric space definition
  4. Open ball definition
  5. Cauchy sequence definition
  6. Complete metric space definition
  7. Completion of a metric space definition
  8. Totally bounded set definition
  9. Compact iff complete and totally bounded statement
  10. Lipschitz condition definition
  11. Isometry definition
  12. Banach fixed point theorem statement
  13. Kuratowski embedding statement
  14. Kirszbraun theorem statement
  15. Polish space definition
  16. Gromov–Hausdorff space object
  17. Urysohn universal space object
  18. Hilbert cube object

Metrization 6 of 6

  1. Metrizable space definition
  2. Urysohn metrization theorem statement
  3. Nagata–Smirnov metrization theorem statement
  4. Bing metrization theorem statement
  5. Moore space definition
  6. Normal Moore space problem problem

Uniform and proximity structures 8 of 8

  1. Uniform space definition
  2. Entourage definition
  3. Uniform continuity definition
  4. Heine–Cantor theorem statement
  5. Cauchy filter definition
  6. Equicontinuity definition
  7. Arzelà–Ascoli theorem statement
  8. Proximity space definition

Continua and dimension theory 12 of 12

  1. Continuum definition
  2. Hahn–Mazurkiewicz theorem statement
  3. Space-filling curve example
  4. Indecomposable continuum definition
  5. Pseudo-arc example
  6. Warsaw circle example
  7. Lakes of Wada example
  8. Knaster–Kuratowski fan counterexample
  9. Lebesgue covering dimension definition
  10. Inductive dimension definition
  11. Sierpiński carpet example
  12. Menger sponge example

Cantor sets and descriptive spaces 5 of 5

  1. Cantor set object
  2. Cantor space object
  3. Brouwer's characterization of the Cantor space statement
  4. Baire space of sequences object
  5. Smith–Volterra–Cantor set example

Ordered spaces and set-theoretic topology 4 of 4

  1. Linear continuum definition
  2. Long line counterexample
  3. Suslin's problem problem
  4. Dowker space definition

Exotic spaces and counterexamples 8 of 8

  1. Sorgenfrey line counterexample
  2. Sorgenfrey plane counterexample
  3. Moore plane counterexample
  4. Line with two origins counterexample
  5. Tychonoff plank counterexample
  6. Arens–Fort space counterexample
  7. Fort space counterexample
  8. Hawaiian earring example

Topology together with algebraic and other structures 7 of 7

  1. Topological group definition
  2. Birkhoff–Kakutani theorem statement
  3. Topological ring definition
  4. Fixed-point property definition
  5. Pointless topology concept
  6. Digital topology concept
  7. Monad in nonstandard topology concept

More ideas in this subject

Concepts 61

Theorems and results 72

Examples 8

Definitions 187

This subject is MSC class 54. See it on zbMATH Open.