Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

The catalog description for Introduction to Topology (MATH 4357/5357) is: "Studies open and closed sets, continuous functions, metric spaces, connectedness, compactness, the real line, and the fundamental group." (ETSU Undergraduate Catalog, 2020-21)

• Chapter 1. Set Theory and Logic.
• Chapter 2. Topological Spaces and Continuous Functions.
• Chapter 3. Connectedness and Compactness.
• Chapter 4. Countability and Separation Axioms.
• Chapter 5. The Tychonoff Theorem.
• Chapter 6. Metrization Theorems and Paracompactness.
• Chapter 7. Complete Metric Spaces and Function Spaces.
• Chapter 8. Baire Spaces and Dimension Theory.

Chapter 1. Set Theory and Logic.

• Section 7. Countable and Uncountable Sets.
• Section 8. The Principle of Recursive Definition.
• Section 9. Infinite Sets and the Axiom of Choice.
• Section 10. Well-Ordered Sets.
• Section 11. The Maximum Principle.
• Study Guide 1.

Chapter 2. Topological Spaces and Continuous Functions.

• Section 12. Topological Spaces. Section 12 notes
• Section 13. Basis for a Topology. Section 13 notes
• Supplement. Proofs of Theorems in Section 13. Beamer file of Section 13 proofs
• Supplement. Printout of the Proofs of Theorems in Section 13. Print file of Section 13 proofs
• Section 14. The Order Topology. Section 14 notes
• Supplement. Proofs of Theorems in Section 14. Beamer file of Section 14 proofs
• Supplement. Printout of the Proofs of Theorems in Section 14. Print file of Section 14 proofs
• Section 15. The Product Topology on X × Y. Section 15 notes
• Supplement. Proofs of Theorems in Section 15. Beamer file of Section 15 proofs
• Supplement. Printout of the Proofs of Theorems in Section 15. Print file of Section 15 proofs
• Section 16. The Subspace Topology. Section 16 notes
• Supplement. Proofs of Theorems in Section 16. Beamer file of Section 16 proofs
• Supplement. Printout of the Proofs of Theorems in Section 16. Print file of Section 16 proofs
• Section 17. Closed Sets and Limit Points. Section 17 notes
• Supplement. Proofs of Theorems in Section 17. Beamer file of Section 17 proofs
• Supplement. Printout of the Proofs of Theorems in Section 17. Print file of Section 17 proofs
• Section 18. Continuous Functions. Section 18 notes
• Supplement. Proofs of Theorems in Section 18. Beamer file of Section 18 proofs
• Supplement. Printout of the Proofs of Theorems in Section 18. Print file of Section 18 proofs
• Section 19. The Product Topology. Section 19 notes
• Supplement. Proofs of Theorems in Section 19. Beamer file of Section 19 proofs
• Supplement. Printout of the Proofs of Theorems in Section 19. Print file of Section 19 proofs
• Section 20. The Metric Topology. Section 20 notes
• Supplement. Proofs of Theorems in Section 20. Beamer file of Section 20 proofs
• Supplement. Printout of the Proofs of Theorems in Section 20. Print file of Section 20 proofs
• Section 21. The Metric Topology (continued). Section 21 notes
• Supplement. Proofs of Theorems in Section 21. Beamer file of Section 21 proofs
• Supplement. Printout of the Proofs of Theorems in Section 21. Print file of Section 21 proofs
• Section 22. The Quotient Topology. Section 22 notes
• Supplement. Proofs of Theorems in Section 22. Beamer file of Section 22 proofs
• Supplement. Printout of the Proofs of Theorems in Section 22. Print file of Section 22 proofs
• Section 22S. Supplement: Topological Groups. Topological Groups notes (This section includes definitions of the general linear group, the special linear group, the orthogonal group, and the special orthogonal group, each over the reals.)
• Study Guide 2.

Chapter 3. Connectedness and Compactness.

• Section 23. Connected Spaces. Section 23 notes
• Supplement. Proofs of Theorems in Section 23. Beamer file of Section 23 proofs
• Supplement. Printout of the Proofs of Theorems in Section 23. Print file of Section 23 proofs
• Section 24. Connected Subspaces of the Real Line. Section 24 notes
• Supplement. Proofs of Theorems in Section 24. Beamer file of Section 24 proofs
• Supplement. Printout of the Proofs of Theorems in Section 24. Print file of Section 24 proofs
• Section 25. Components and Local Connectedness. Section 24 notes
• Supplement. Proofs of Theorems in Section 25. Beamer file of Section 25 proofs
• Supplement. Printout of the Proofs of Theorems in Section 25. Print file of Section 25 proofs
• Section 26. Compact Spaces. Section 26 notes
• Supplement. Proofs of Theorems in Section 26. Beamer file of Section 26 proofs
• Supplement. Printout of the Proofs of Theorems in Section 26. Print file of Section 26 proofs
• Section 27. Compact Subspaces of the Real Line. Section 27 notes
• Supplement. Proofs of Theorems in Section 27. Beamer file of Section 27 proofs
• Supplement. Printout of the Proofs of Theorems in Section 27. Print file of Section 27 proofs
• Section 28. Limit Point Compactness. Section 28 notes
• Supplement. Proofs of Theorems in Section 28. Beamer file of Section 28 proofs
• Supplement. Printout of the Proofs of Theorems in Section 28. Print file of Section 28 proofs
• Section 29. Local Compactness. Section 29 notes
• Supplement. Proofs of Theorems in Section 29. Beamer file of Section 29 proofs
• Supplement. Printout of the Proofs of Theorems in Section 29. Print file of Section 29 proofs
• Study Guide 3.

Chapter 4. Countability and Separation Axioms. PDF.

• Section 30. The Countability Axioms. Section 30 notes
• Supplement. Proofs of Theorems in Section 30. Beamer file of Section 30 proofs
• Supplement. Printout of the Proofs of Theorems in Section 30. Print file of Section 30 proofs
• Section 31. The Separation Axioms. Section 31 notes
• Supplement. Proofs of Theorems in Section 31. Beamer file of Section 31 proofs
• Supplement. Printout of the Proofs of Theorems in Section 31. Print file of Section 31 proofs
• Section 32. Normal Spaces. Section 32 notes
• Supplement. Proofs of Theorems in Section 32. Beamer file of Section 32 proofs
• Supplement. Printout of the Proofs of Theorems in Section 32. Print file of Section 32 proofs
• Section 33. The Urysohn Lemma. Section 33 notes
• Supplement. Proofs of Theorems in Section 33. Beamer file of Section 33 proofs
• Supplement. Printout of the Proofs of Theorems in Section 33. Print file of Section 33 proofs
• Section 34. The Urysohn Metrization Theorem. Section 34 notes
• Supplement. Proofs of Theorems in Section 34. Beamer file of Section 34 proofs
• Supplement. Printout of the Proofs of Theorems in Section 34. Print file of Section 34 proofs
• Section 35. The Tietze Extension Theorem. Section 35 notes
• Supplement. Proofs of Theorems in Section 35. Beamer file of Section 35 proofs
• Supplement. Printout of the Proofs of Theorems in Section 35. Print file of Section 35 proofs
• Section 36. Imbeddings of Manifolds. Section 36 notes
• Supplement. Proofs of Theorems in Section 36. Beamer file of Section 36 proofs
• Supplement. Printout of the Proofs of Theorems in Section 36. Print file of Section 36 proofs
• Study Guide 4.

Chapter 5. The Tychonoff Theorem.

• Section 37. The Tychonoff Theorem. Section 37 notes
• Supplement. Proofs of Theorems in Section 37. Beamer file of Section 37 proofs
• Supplement. Printout of the Proofs of Theorems in Section 37. Print file of Section 37 proofs
• Section 38. The Stone-Cech Compactification.
• Study Guide 5.

Chapter 6. Metrization Theorems and Paracompactness.

• Section 39. Local Finiteness. Section 39 notes
• Supplement. Proofs of Theorems in Section 39. Beamer file of Section 39 proofs
• Supplement. Printout of the Proofs of Theorems in Section 39. Print file of Section 39 proofs
• Section 40. The Nagata-Smirnov Metrization Theorem. Section 40 notes
• Supplement. Proofs of Theorems in Section 40. Beamer file of Section 40 proofs
• Supplement. Printout of the Proofs of Theorems in Section 40. Print file of Section 40 proofs
• Section 41. Paracompactness. Section 41 notes
• Supplement. Proofs of Theorems in Section 41. Beamer file of Section 41 proofs
• Supplement. Printout of the Proofs of Theorems in Section 41. Print file of Section 41 proofs
• Section 42. The Smirnov Metrization Theorem. Section 42 notes
• Supplement. Proofs of Theorems in Section 42. Beamer file of Section 42 proofs
• Supplement. Printout of the Proofs of Theorems in Section 42. Print file of Section 42 proofs
• Study Guide 6.

Chapter 7. Complete Metric Spaces and Function Spaces.

• Section 43. Complete Metric Spaces. Section 43 notes (This section includes a discussion of Cauchy sequences in a topological space.)
• Supplement. Proofs of Theorems in Section 43. Beamer file of Section 43 proofs
• Supplement. Printout of the Proofs of Theorems in Section 43. Print file of Section 43 proofs
• Section 44. A Space-Filling Curve. Section 44 notes
• Section 45. Compactness in Metric Spaces. Section 45 notes
• Supplement. Proofs of Theorems in Section 45. Beamer file of Section 45 proofs
• Supplement. Printout of the Proofs of Theorems in Section 45. Print file of Section 45 proofs
• Section 46. Pointwise and Compact Convergence.
• Section 47. Ascoli's Theorem.
• Study Guide 7.

Chapter 8. Baire Spaces and Dimension Theory.

• Section 48. Baire Spaces.
• Section 49. A Nowhere-Differentiable Function.
• Section 50. Introduction to Dimension Theory.
• Study Guide 8.

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