Content: Topological spaces, Continuous maps between topological spaces, product topology, Quotient spaces, Connectedness, Compactness, Path connected spaces, separation axioms, Tychonov spaces, Urysohn's lemma and metrization theorem. Differentiable functions on \(\mathbb{R}^n\), Jacobian criteria, Taylor's theorem, Inverse function theorem, Implicit function theorem, Maxima-minima, Lagrange multiplier.

Texts:

• James R. Munkres, "Topology", Pearson, 2nd edition, 2005.
• Michael Spivak, “Calculus on Manifolds”, Springer Verlag, 2nd edition, 2013.
• James R. Munkres, "Analysis on Manifolds", CRC Press, Reprint 2022.
• R. Gunning, "Introduction to Functions of Several Real Variables", Course Notes.

References:

• Robert André, "Point-Set Topology with Topics: Basic General Topology for Graduate Studies", World Scientific Publishing, 2024.
• Kelley, John L., “General Topology”, Grad. Texts in Math., No. 27, Springer-Verlag, 1975.
• James Dugundji, "Topology", Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, 1978.
• Stephen Willard, "General topology", Addison-Wesley Publishing Co. 1970.
• Steen and Seebach, "Counterexamples in topology", Second edition, Dover Publications, 1995.
• B. B. Hubbard, J. H. Hubbard, "Vector calculus, linear algebra, and differential forms" Prentice Hall, 1999.
• Theodore Shifrin, "Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds", Wiley, 2004.

Other resources: Lectures by Theodore Shifrin

Evaluation: Problem sets 30%, Midsem 30%, Final 40%.
Exams: To be announced later.