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Topology: Handwritten Notes

A handwritten notes of Topology by Mr. Tahir Mehmood. These notes covers almost every topic which required to learn for MSc mathematics. House of Tau

Name Topology: Handwritten Notes
Author Mr. Tahir Mahmood
Pages 262 pages
Format Scanned PDF
Size 10.08 mB

Contents & Summary

  • Metric space … 1
  • Minkowski's inequality … 5
  • Open set … 7
  • Closed ball … 9
  • Closed set … 10
  • Bounded set … 11
  • Limit point … 13
  • Closure of a set … 14
  • Convergence in metric space and complete metric space … 18
  • Cauchy sequence … 19
  • Bounded sequence … 20
  • Nested interval property or Cantor's intersection theorem … 26
  • Continuous function … 28
  • Topological spaces … 38
  • Metric topology, cofinite topology … 39
  • Open set … 41
  • Closed set … 43
  • Closure of a set … 44
  • Neighbourhood … 48
  • Interior point, exterior point … 49
  • Boundary point … 50
  • Limit point (with respect to topology) … 52
  • Isolated point … 62
  • Dense … 63
  • Separable set; Countable set … 64
  • Base of topology … 65
  • Neighbourhood base or local base or base at a point … 71
  • Open cover; Lindelof space … 73
  • Lindelof theorem … 74
  • Relative topology, subspace … 77
  • Separation axioms; $T_0$-space … 85
  • $T_1$-space … 87
  • Subbase; Generation of topologies … 92
  • $T_2$-space … 93
  • Continuous function (with respect to topologies) … 95
  • Product topology … 98
  • Convergence of sequence in topological spaces … 101
  • Regular space … 109
  • Completely regular space … 111
  • Compactness in topological spaces … 125
  • Homeomorphism … 134
  • Countably compact space … 141
  • Bolzano Weierstrass property … 145
  • Lebesgue number; Big set; Lebesgue covery lemma … 147
  • $\varepsilon-$net; Totally bounded … 149
  • Connected spaces; Disconnected … 157
  • Component … 170
  • Totally disconnected … 173
  • Separated … 180
  • Normed spaced … 186
  • Uniformly continuous … 189
  • Closed unit ball; Convex set … 190
  • Vector space … 191
  • Linear combination; Spanning set; Linearly independent … 192
  • Linearly dependent … 193
  • Linearly independent lemma … 194
  • Finite dimensional; Subspace … 197
  • Equivalent norms … 200
  • Banach space … 205
  • Reiz Lemma … 222
  • Hilbert spaces; Inner product spaces … 224
  • Polarization identity … 228
  • Cauchy Schewarz inequality … 229
  • Appalonius identity … 231
  • Hilbert space; Pythagorian theorem … 233
  • Minimizing vector … 235
  • Direct sum … 238
  • Orthogonal set; Orthonormal set … 242
  • Bessel's inequality … 243
  • Total orthonormal sets (definition); Parsevel's equality … 245
  • Linear Operator; The Kernel or Null space of a linear operator; Continuous linear operator … 247
  • Bounded linear operator … 249
  • Norm of a bounded lienar operator … 252
  • Linear functionals … 260

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