Introduction to topology

18.901: Introduction to topology (Spring 2025)

The syllabus for this course is available here.

Lecture time: Tuesday and Thursday: 1:00 pm - 2:30 pm

Location: 2-190

Instructor Email: pieloch@mit.edu
Instructor Office Hours: TBD


Problem Sets

  • Problem set 5
    This assignment is not due, will not be graded, and should not be turned in.

  • Problem set 10
    This assignment is not due, will not be graded, and should not be turned in.





Exams

  • Midterm 1:
  • Midterm 2:
  • Final exam:
    • Time: Wednesday, May 21st, 2025 at 9:00 am - 12:00 pm.
    • Material covered: All material that was covered in the lectures during the entirety of the course.




Course Schedule

This section will be updated after class to list the material that was covered along with the lecture notes. For a tentative schedule of the course, see the syllabus.

  • Week 1
    • February 4th, 2025:
      • Introduction, topological spaces, bases, interiors, closures, limit points
      • Lecture notes: 1.1, 1.2
    • February 6th, 2025:
      • Dense subsets, metric spaces, subspaces, product spaces
      • Lecture notes: 1.2, 1.3

  • Week 2
    • February 11th, 2025:
      • Quotient topology, continuous maps
      • Lecture notes: 1.3.4, 1.4
    • February 13th, 2025:
      • Continuous maps, limit points
      • Lecture notes: 1.4

  • Week 3
    • February 18th, 2025:
      • No class
    • February 20th, 2025:
      • Connectedness
      • Lecture notes: 1.5

  • Week 4
    • February 25th, 2025:
      • Compactness, Hausdorff spaces
      • Lecture notes: 1.6, 1.7
    • February 20th, 2025:
      • Normal spaces
      • Lecture notes: 1.8.1

  • Week 5
    • March 4th, 2025:
      • Urysohn's lemma, Urysohn's metrization theorem
      • Lecture notes: 1.8.2
    • March 6th, 2025:
      • Midterm 1

  • Week 6
    • March 11th, 2025:
      • Manifolds
      • Lecture notes: 2.1
    • March 13th, 2025:
      • Paracompactness
      • Lecture notes: 2.2

  • Week 7
    • March 18th, 2025:
      • Covering dimension, metric space embedding theorem
      • Lecture notes: 2.3, 2.4
    • March 20th, 2025:
      • Manifold embedding theorem, Homotopy
      • Lecture notes: 2.4, 3.1

  • Week 8
    • March 25th, 2025:
      • No class
    • March 27th, 2025:
      • No class

  • Week 9
    • April 1st, 2025:
      • Groups, subgroups, homomorphisms
      • Lecture notes: 4.1.1, 4.1.2, 4.1.3
    • April 3rd, 2025:
      • Normal subgroups, quotient groups, definition of fundamental group
      • Lecture notes: 4.1.4, 3.2

  • Week 10
    • April 8th, 2025:
      • Change-of-basepoint, induced maps, fibre bundles
      • Lecture notes: 3.3.1, 3.3.2, 3.4.1
    • April 10th, 2025:
      • More fibre bundles, the homotopy lifting property, computations of fundamental groups
      • Lecture notes: 3.4.1, 3.4.2

  • Week 11
    • April 17th, 2025:
      • Applications of fundamental groups, free groups, presentations of groups, amalgamated free products
      • Lecture notes: 3.5, 4.2
    • April 10th, 2025:
      • Midterm 2

  • Week 12
    • April 22th, 2025:
      • The van Kampen theorem, fundamental groups of graphs
      • Lecture notes: 3.6.1, 3.6.2
    • April 24th, 2025:
      • Fundamental groups of surfaces, simplicial complexes
      • Lecture notes: 3.6.2, 4.1.1, 4.1.2, 4.1.3, 4.1.4

  • Week 13
    • April 29th, 2025:
      • More simplicial complexes, simplicial approximation theorem, quotient vector spaces
      • Lecture notes: 4.2.1, 4.2.2, 5.3.2
    • May 1st, 2025:
      • Chain complexes, chain maps, chain homotopies, exact sequences, the snake lemma
      • Lecture notes: 5.4.1, 5.4.2, 5.4.3, 5.4.4

  • Week 14
    • May 6th, 2025:
      • Simplicial homology, homology computations
      • Lecture notes:
    • May 8th, 2025:
      • Invariance of simplicial homology
      • Lecture notes:

  • Week 15
    • May 13th, 2025:
      • Applications of homology.
      • Lecture notes: