Topology I

Semester I (PS01CMTH52)

Syllabus

Unit-I Topological spaces, basis, subbasis, the product topology on $X \times Y$, the subspace topology, closed sets, closure and interior, limit points, boundary of a set.
Unit-II Hausdorff spaces, convergent sequence, $T_1$-space, Continuous functions, homeomorphisms, constructing continuous functions, pasting lemma, metric topology, diameter and bounded sets, bounded metric $\overline{d}$ (excluding norm), continuity in metrizable spaces, the sequence lemma, first countability axiom.
Unit-III Connected spaces, connected subspaces of the real line, connected components, compact spaces, finite intersection property, Heine-Borel theorem for real line, second countable spaces, separable spaces.
Unit-IV                     Regular spaces, Normal spaces, Urysohn’s Lemma (statement only), Tietze’s Extension Theorem (statement only), Complete metric spaces, Cantor’s intersection theorem, Baire’s category theorem for complete metric spaces.


Reference Books

  1. Munkres J., Topology: A First Course, (Second Edition), Prentice Hall of India Pvt. Ltd. New Delhi, 2003.
  2. Simmons G.F., Introduction to Topology and Modern Analysis, McGraw-Hill Co., Tokyo, 1963.
  3. Willard S., General Topology, Dover Publication, 2004.
  4. Kelley J., General Topology, Graduate Texts in Mathematics, Springer-Verlag, 1975.

Lecture Notes