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
  • Minkowski's inequality
  • Open set
  • Closed ball
  • Closed set
  • Bounded set
  • Limit point
  • Closure of a set
  • Convergence in metric space and complete metric space
  • Cauchy sequence
  • Bounded sequence
  • Nested interval property or Cantor's intersection theorem
  • Continuous function
  • Topological spaces
  • Metric topology, cofinite topology
  • Open set
  • Closed set
  • Closure of a set
  • Neighbourhood
  • Interior point, exterior point
  • Boundary point
  • Limit point (with respect to topology)
  • Isolated point
  • Dense
  • Separable set; Countable set
  • Base of topology
  • Neighbourhood base or local base or base at a point
  • Open cover; Lindelof space
  • Lindelof theorem
  • Relative topology, subspace
  • Separation axioms; $T_0$-space
  • $T_1$-space
  • Subbase; Generation of topologies
  • $T_2$-space
  • Continuous function (with respect to topologies)
  • Product topology
  • Convergence of sequence in topological spaces
  • Regular space
  • Completely regular space
  • Compactness in topological spaces
  • Homeomorphism
  • Countably compact space
  • Bolzano Weierstrass property
  • Lebesgue number; Big set; Lebesgue covery lemma
  • $\varepsilon-$net; Totally bounded
  • Connected spaces; Disconnected
  • Component
  • Totally disconnected
  • Separated
  • Normed spaced
  • Uniformly continuous
  • Closed unit ball; Convex set
  • Vector space
  • Linear combination; Spanning set; Linearly independent
  • Linearly dependent
  • Linearly independent lemma
  • Finite dimensional; Subspace
  • Equivalent norms
  • Banach space
  • Reiz Lemma
  • Hilbert spaces; Inner product spaces
  • Polarization identity
  • Cauchy Schewarz inequality
  • Appalonius identity
  • Hilbert space; Pythagorian theorem
  • Minimizing vector
  • Direct sum
  • Orthogonal set; Orthonormal set
  • Bessel's inequality
  • Total orthonormal sets (definition); Parsevel's equality
  • Linear Operator; The Kernel or Null space of a linear operator; Continuous linear operator
  • Bounded linear operator
  • Norm of a bounded lienar operator
  • Linear functionals

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