Math 2.0

Learn Year 2

Metric spaces and topology

170 ideas in teaching order. Each idea comes after the ideas that it builds on.

Start with Metric space →

  1. Metric space definition
  2. Triangle inequality for a metric statement
  3. Pseudometric space definition
  4. Usual metric on the real line example
  5. Euclidean metric example
  6. Taxicab metric example
  7. Sup metric on R^n example
  8. p-metric on R^n example
  9. Discrete metric example
  10. Uniform metric on bounded functions example
  11. Subspace metric definition
  12. Product metric definition
  13. Open ball definition
  14. Closed ball definition
  15. Bounded set definition
  16. Diameter of a set definition
  17. Distance from a point to a set definition
  18. Open set definition
  19. Open sets of a metric space statement
  20. Open balls are open statement
  21. Union of open sets is open statement
  22. Finite intersection of open sets is open statement
  23. Closed set definition
  24. Complement of an open set is closed statement
  25. Closed balls are closed statement
  26. Intersection of closed sets is closed statement
  27. Interior definition
  28. Interior is open statement
  29. Interior is the largest open subset statement
  30. Closure definition
  31. Closure is closed statement
  32. Closure is the smallest closed superset statement
  33. A set is closed iff it equals its closure statement
  34. Closure in a metric space statement
  35. Limit point definition
  36. Isolated point definition
  37. Boundary definition
  38. Dense set definition
  39. Separable space definition
  40. Convergence of a sequence in a metric space definition
  41. Closed sets contain their limits statement
  42. Closure is the set of limits of sequences statement
  43. Epsilon-delta continuity in metric spaces statement
  44. Continuity is equivalent to sequential continuity in metric spaces statement
  45. Continuity via open sets statement
  46. Continuity via closed sets statement
  47. Composition of continuous maps is continuous statement
  48. Epsilon-delta uniform continuity statement
  49. Lipschitz map definition
  50. Lipschitz maps are uniformly continuous statement
  51. Hölder continuity definition
  52. Isometry definition
  53. Isometries are injective statement
  54. Isometric isomorphism definition
  55. Equivalent metrics definition
  56. Epsilon characterisation of Cauchy sequences statement
  57. Complete metric space definition
  58. Completeness via Cauchy sequences statement
  59. Closed subsets of complete spaces are complete statement
  60. Complete subspaces are closed statement
  61. Completeness of the space of bounded continuous functions statement
  62. Completion of a metric space definition
  63. Contraction mapping definition
  64. Banach fixed point theorem statement
  65. Cantor's intersection theorem statement
  66. Nowhere dense set definition
  67. Meagre set definition
  68. Baire space definition
  69. Baire category theorem statement
  70. Countable intersection of dense open sets is dense statement
  71. Open cover definition
  72. Compact set definition
  73. Compactness via open covers statement
  74. Compact space definition
  75. Finite intersection property statement
  76. Sequential compactness definition
  77. Compactness and sequential compactness agree in metric spaces statement
  78. Compact subsets of a metric space are bounded statement
  79. Compact subsets of a Hausdorff space are closed statement
  80. Closed subsets of compact sets are compact statement
  81. Heine–Borel theorem statement
  82. Proper metric space definition
  83. Continuous image of a compact set is compact statement
  84. Continuous real function on a compact set attains its bounds statement
  85. Heine–Cantor theorem statement
  86. Lebesgue number lemma statement
  87. Totally bounded set definition
  88. Epsilon-net characterisation of total boundedness statement
  89. Compact iff complete and totally bounded statement
  90. Product of compact sets is compact statement
  91. Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
  92. Equicontinuity definition
  93. Arzelà–Ascoli theorem statement
  94. Topological space definition
  95. Open sets characterised by neighbourhoods statement
  96. Metric topology definition
  97. Discrete topology example
  98. Indiscrete topology example
  99. Cofinite topology example
  100. Metrizable space definition
  101. Comparison of topologies definition
  102. Basis of a topology definition
  103. Open balls form a basis of the metric topology statement
  104. Subbasis definition
  105. First-countable space definition
  106. Second-countable space definition
  107. Continuous map of topological spaces definition
  108. Continuity at a point of a topological space definition
  109. Homeomorphism definition
  110. Topological property concept
  111. Open map definition
  112. Closed map definition
  113. Subspace topology definition
  114. Inclusion of a subspace is continuous statement
  115. Topological embedding definition
  116. Product topology definition
  117. Projections are continuous statement
  118. Product topology on an arbitrary product definition
  119. Universal property of the product topology statement
  120. Box topology definition
  121. Quotient topology definition
  122. Universal property of the quotient topology statement
  123. Circle as a quotient of the real line example
  124. Torus example
  125. Möbius strip example
  126. Separation axioms concept
  127. T0 space definition
  128. T1 space definition
  129. Hausdorff space definition
  130. Limits are unique in Hausdorff spaces statement
  131. Metric spaces are Hausdorff statement
  132. Regular space definition
  133. Normal space definition
  134. Metrizable spaces are normal statement
  135. Compact Hausdorff spaces are normal statement
  136. Urysohn's lemma statement
  137. Tietze extension theorem statement
  138. Tychonoff's theorem statement
  139. Locally compact space definition
  140. One-point compactification definition
  141. Connected space definition
  142. Connected set definition
  143. Connected subsets of the real line are intervals statement
  144. Continuous image of a connected set is connected statement
  145. Intermediate value theorem for connected spaces statement
  146. Path-connected space definition
  147. Path-connected spaces are connected statement
  148. Open connected subsets of R^n are path-connected statement
  149. Connected components are closed statement
  150. Path component definition
  151. Totally disconnected space definition
  152. Topologist's sine curve counterexample
  153. Cantor set example
  154. Cantor set is closed statement
  155. Cantor set is homeomorphic to the space of binary sequences statement
  156. p-norm on R^n example
  157. Sup norm on bounded continuous functions example
  158. Sequence spaces lp example
  159. Equivalent norms definition
  160. Equivalence of norms in finite dimension statement
  161. Banach space definition
  162. Finite-dimensional normed spaces are complete statement
  163. Heine–Borel theorem in finite-dimensional normed spaces statement
  164. Absolutely convergent series in a Banach space converge statement
  165. Continuous linear map definition
  166. Operator norm definition
  167. Bounded linear maps are continuous statement
  168. Linear maps on finite-dimensional spaces are continuous statement
  169. Riesz's lemma statement
  170. Compact closed unit ball implies finite dimension statement