Lecture 1: Metric spaces and their properties 
PDF 
Lecture 2: Closed sets and sequences 
PDF 
Lecture 3: Cauchy sequences and completeness 
PDF 
Lecture 4: Baire category theorem and completeness of $\mathbb R$ 
PDF 
Lecture 5: Continuity 
PDF 
Lecture 6: Products 
PDF 
Lecture 7: Compactness 
PDF 
Lecture 8: Compactness in metric spaces 
PDF 
Lecture 9: Metric compactness continued 
PDF 
Lecture 10: Compactness and normed spaces 
PDF 
Lecture 12: Topological spaces 
PDF 
Lecture 13: Continuous functions and subspaces 
PDF 
Lecture 14: Bases and subbases 
PDF 
Lecture 15: Products and quotients 
PDF 
Lecture 16: Separation axioms 
PDF 
Lecture 17: $T_3$ and $T_4$ 
PDF 
Lecture 18: Uryshon and Tietze 
PDF 
Lecture 19: Compactness 
PDF 
Lecture 20: Connectedness 
PDF 
Lecture 21: Locally compact spaces and infinite products 
PDF 
Lecture 23: Groups and homotopies 
PDF 
Lecture 24: Paths and the fundamental group 
PDF 
Lecture 25: Induced homomorphisms and base points 
PDF 
Lecture 26: Covering spaces 

Lecture 27: Path lifting 
PDF 
Lecture 28: Covers and $\pi_1$ 
PDF 
Lecture 29: Classification 
PDF 