ONE METRIC SPACES 1 Open and closed sets2 Completeness3 The real line4 Products of metric spaces5 Compactness6 Continuous functions7 Normed linear spaces8 The contraction principle9 The Frechet derivative TWO TOPOLOGICAL SPACES1 Topological spaces2 Subspaces3 Continuous functions4 Base for a topology5 Separation axioms6 Compactness7 Locally compact spaces8 Connectedness9 Path connectedness10 Finite product spaces11 Set theory and Zorn's lemma12 Infinite product spaces13 Quotient spaces THREE HOMOTOPY THEORY1 Groups2 Homotopic paths3 The fundamental group4 Induced homomorphisms5 Covering spaces6 Some applications of the index 7 Homotopic maps8 Maps into the punctured plane9 Vector fields10 The Jordan Curve Theorem FOUR HIGHER DIMENSIONAL HOMOTOPY 1 Higher homotopy groups2 Noncontractibility of Sn 3 Simplexes and barycentric subdivision 4 Approximation by piecewise linear maps 5 Degrees of maps BIBLIOGRAPHY LIST OF NOTATIONS SOLUTIONS TO SELECTED EXERCISES INDEX