Main reference books for the course:

• General Topolgy, Volumes 1 & 2 , by N. Bourbaki
• General Topolgy, byJ. L. Kelley
• Topology 2nd Edition, by James R. Munkres, which is available in Indian paperback edition

To become a member of the google groups for the course, send an email to the instructor, K N Raghavan, at knr at imsc dot res dot in.

The course meets in Room 217 on Mondays, Wednesdays, and Fridays during 930--1100 hrs.

Notes:

• Chapter I: Topological Structures (following Chapter I of Bourbaki)
  • Section 1: Neighborhoods
    • I.1: par 1 & 2   open sets, neighborhoods, closed sets
    • I.1: par 3 to 5   fundamental systems of neighborhoods, bases, closed sets, locally finite families
    • I.1: par 6   interior, frontier, and exterior; dense sets
    • I.1   summary
    • Exercises for section 1:   page 1,   page 2
  • Section 2: Continuous functions
    • I.2 par 1   continuous functions
    • I.2 par 2 and 3   comparison of topologies; initial topologies
    • I.2 par 4   final topologies
    • I.2 par 5   pasting together of topological spaces
  • Section 3: Subspaces and quotient spaces
    • I.3 par 1   Subspaces
    • I.3 par 2 & 3   Continuity with respect to a subspace; Locally closed subspaces
    • I.3 par 4   Quotient spaces
    • I.3 par 5   Canonical decomposition of a continuous mapping
    • I.3 par 6   Quotient space of a subspace
  • Section 4: Products of topological spaces
    • I.4 par 1   Product spaces
    • I.4 par 2   Sections, projections, and partial continuity
    • I.4 par 3 & 4   Closures in products, inverse limits
  • Section 5: Open maps and closed maps
    • I.5 par 1   Open maps and closed maps
    • I.5 par 2   Open (resp. closed) equivalence relations
    • I.5 par 3   Properties peculiar to open mappings
  • Section 6: Filters
    • I.6 par 1 & 2   Filters, comparison of filters
    • I.6 par 3   Filter bases
    • I.6 par 4   Ultrafilters
    • I.6 par 5 & 6   Induced filters, direct and inverse images of filters
    • I.6 par 6   Direct and inverse images of filters (continued)
    • I.6 par 7 and 8   Product of filters, elementary filters
    • I.6 par 9   Germs with respect to a filter
    • I.6 par 10   Germs at a point
    • Exercises for section 6:   page 1 , page 2
  • Section 7: Limits
    • I.7 par 1   Limits of a filter
    • I.7 par 2 and 3   Cluster points of a filter base, limits and cluster values of a function
    • I.7 par 3 (contd.) and 4   Limits and cluster values of a function, limits and continuity
    • I.7 par 4 (contd.), 5, & 6   Limits relative to a subspace; limits in initial topologies, products, and quotients
    • Exercises for section 7:  page 1
  • Section 8: Hausdorff spaces and regular spaces
    • I.8 par 1   Hausdorff spaces
    • I.8 par 2   Subspaces and products of Hausdorff spaces
    • I.8 par 3   Hausdorff quotient spaces
    • I.8 par 4   Regular spaces
    • I.8 par 5   Extension by continuity; Double limit
    • I.8 par 6   Equivalence relations on regular spaces
  • Section 9: Compact spaces and locally compact spaces
    • I.9 par 1   Quasi-compact spaces and compact spaces
    • I.9 par 2 & 3   Regularity of a compact space; quasi-compact sets, compact sets, and relatively compact sets
    • I.9 par 3   Relative compactness (continued)
    • I.9 par 4 and 5   Image of a compact space under a continuous mapping; products of compact spaces (Tychonoff's theorem)
    • I.9 par 6   Inverse limits of compact spaces
    • I.9 par 7   Locally compact spaces
    • I.9 par 8   One-point (or Alexandroff) compactification
    • I.9 par 9   Locally compact and sigma-compact spaces
    • I.9 par 10   Paracompact spaces
    • I.9 par 10 (Continued)   Structure theorem for locally compact paracompact spaces
  • Miscaellaneous Exercises:   page 1
• Chapter II: Uniform structures (following Chapter 6 of Kelley)
  • Uniformities and uniform topologies
    • Uniform topology; bases and subbases for uniformities
    • Open (resp. closed) symmetric entourages form a base; accessible uniform spaces are regular
    • Uniform continuity; initial uniformities: subspace and product uniformities
    • Metrization lemma and metrization theorem
    • Proof of metrization lemma
    • Uniformities and pseudo-metrics
  • Completions
    • Completeness of uniform spaces
    • Products of complete spaces
    • Products of complete spaces (continued)
    • Completions; Compact uniform spaces
  • For metric spaces only
    • Baire category theorem
    • A theorem about meagerness
    • Uniformly open maps; statement without proof: the image in a Hausdorff uniform space of a complete pseudo-metric space under a uniformly open continuous map is complete
  • Function spaces
    • Function spaces
    • Topology of pointwise convergence
    • Pointwise uniformity
    • Compact open topology
• Chapter III: Fundamental group and covering spaces (following Munkres Chapter 8)
  • Homotopies, path homotopies, definition of the fundamental group
  • Covering spaces
  • Fundamental group of the circle
  • retract and strong deformation retract; fundamental groups of spheres (special van Kampen theorem)
• Chapter III-A: Covering spaces and the principle of monodromy (following Chevalley Chapter II Section VI)
  • Covering spaces: locally connected spaces, evenly covered sets
  • Covering spaces: some lemmas
  • Simply connected spaces
  • The principle of mondromy
• Appendix: z-filters on a topological space X and ideals in the ring C(X) of real-valued continuous functions on X (following Gillman & Jerison)
• zero sets, z-filters and ideals in C(X), z-ultra filters and maximal ideals in C(X)
• z-ideals, prime z-filters and prime ideals in C(X)
• On minimal prime ideals in C(X)