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 1
Due Thursday, February 6th, 2025 at 11:59 am.
- Problem set 2
Due Thursday, February 13th, 2025 at 11:59 am.
- Problem set 3
Due Thursday, February 20th, 2025 at 11:59 am.
- Problem set 4
Due Thursday, February 27th, 2025 at 11:59 am.
- Problem set 5
This assignment is not due, will not be graded, and should not be turned in.
- Problem set 6
Due Thursday, March 13th, 2025 at 11:59 am.
- Problem set 7
Due Thursday, March 20th, 2025 at 11:59 am.
- Problem set 8
Due Thursday, April 3rd, 2025 at 11:59 am.
- Problem set 9
Due Thursday, April 10th, 2025 at 11:59 am.
- Problem set 10
This assignment is not due, will not be graded, and should not be turned in.
- Problem set 11
Due Thursday, April 24th, 2025 at 11:59 am.
- Problem set 12
Due Thursday, May 1st, 2025 at 11:59 am.
Exams
- Midterm 1:
- Time: Thursday, March 6th, 2025 at 1:00 pm - 2:30 pm
- Material covered: Lectures 1-8
- Blank Midterm 1, Solutions to Midterm 1
- Midterm 2:
- Time: Thursday, April 17th, 2025 at 1:00 pm - 2:30 pm
- Material covered: All material from Lecture 9 up to (but no including) free groups.
- Blank Midterm 2, Solutions to 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
- February 4th, 2025:
- 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
- February 11th, 2025:
- Week 3
- February 18th, 2025:
- No class
- February 20th, 2025:
- Connectedness
- Lecture notes: 1.5
- February 18th, 2025:
- Week 4
- February 25th, 2025:
- Compactness, Hausdorff spaces
- Lecture notes: 1.6, 1.7
- February 20th, 2025:
- Normal spaces
- Lecture notes: 1.8.1
- February 25th, 2025:
- Week 5
- March 4th, 2025:
- Urysohn's lemma, Urysohn's metrization theorem
- Lecture notes: 1.8.2
- March 6th, 2025:
- Midterm 1
- March 4th, 2025:
- Week 6
- March 11th, 2025:
- Manifolds
- Lecture notes: 2.1
- March 13th, 2025:
- Paracompactness
- Lecture notes: 2.2
- March 11th, 2025:
- 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
- March 18th, 2025:
- Week 8
- March 25th, 2025:
- No class
- March 27th, 2025:
- No class
- March 25th, 2025:
- 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
- April 1st, 2025:
- 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
- April 8th, 2025:
- 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
- April 17th, 2025:
- 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
- April 22th, 2025:
- 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
- April 29th, 2025:
- Week 14
- May 6th, 2025:
- Simplicial homology, homology computations
- Lecture notes:
- May 8th, 2025:
- Invariance of simplicial homology
- Lecture notes:
- May 6th, 2025:
- Week 15
- May 13th, 2025:
- Applications of homology.
- Lecture notes:
- May 13th, 2025: