18.S190 | January IAP 2023 | Undergraduate

Introduction to Metric Spaces

Instructor Insights

Instructor Interview

Below, Paige Bright describes various aspects of how she taught 18.S190 Introduction to Metric Spaces during the 2023 Independent Activities Period (IAP).

OCW: For the uninitiated, can you explain for us what metric spaces are and why they’re an interesting topic to study?

Paige Bright: Metric spaces allow us to rigorously study distance. Up to this point in math, we often want things to look perfect, using the Pythagorean theorem to understand distances in n-dimensional space. However, even in our day-to-day life this isn’t how we understand space. Consider for instance, a city with a grid system like New York. If I wanted to walk from one block to another, the distance wouldn’t be the shortest way to get there (after all, I cannot walk through walls). Rather, it would be the number of blocks I walked (times the distance of the block). So while we have a good understanding of distances in Euclidean space (as is studied in 18.100A Introduction to Analysis and 18.100P Real Analysis), the question in turn becomes “how does this generalize?” —a question this class attempts to begin to answer.

OCW: You first taught this class in 2022 when you yourself were a second-year undergraduate. How did you wind up in this unusual position?

Paige Bright: When taking 18.102 Introduction to Functional Analysis with Dr. Casey Rodriguez, I realized there was a small but important gap between 18.100A [real analysis on Euclidean space] and 18.100B [real analysis in more abstract settings]: metric spaces. I firmly believe there are many great reasons to take 18.100A, and felt that students should have resources to fill this gap for later courses if they wanted. I realized that this resource didn’t exist after I saw many friends struggling with missing familiarity with abstract spaces. Thus, I started advocating for this course, which gained traction with analysis faculty. I reached out to Academic Administrator Barbara Peskin to figure out the actual logistics of running a class, who put me in contact with Professor Bill Minicozzi, Associate Department Head for Mathematics, who helped me develop a syllabus. A few months down the road I put together the material to teach the class into existence!

OCW: Was this one of your first formal teaching experiences? In developing the course, how much thought did you give to your teaching technique, as opposed to the material you planned to cover?

Paige Bright: I have known since second grade that I have wanted to teach, though this is my first time being the actual teacher. Up until this point, in high school and while at MIT, I have tutored for numerous courses from trigonometry and beyond. When developing the course, I spent a lot of time fine-tuning the material pedagogically. My goal was to make it clear where this subject goes from here, motivating classes like 18.901 Introduction to Topology, 18.101 Analysis and Manifolds, 18.103 Introduction to Fourier Analysis, and 18.102 Functional Analysis. Ultimately, the material in Lebl’s chapter on metric spaces is only 30 pages long, but the power of this tool often goes unexplained, something I wanted to fix in this version of the class. I was also highly influenced by the teachers and students around me who have fostered amazing learning environments. These people influenced me to create an accessible website for students, type lecture notes, and record lectures during a peak in COVID.

OCW: Last year you participated in a conference session on student participation in creating open educational resources. Can you share your vision of the role students could or should play in the educational ecosystem?

Paige Bright: I wish students took more of a part in the classroom, and I believe this can be better encouraged by professors. The most influential courses at MIT in the math department, in my opinion, are those with course material available to the public after the fact. For instance, when Professor Larry Guth, the faculty advisor for both versions of this course, taught the course 18.118 Topics in Analysis: Decoupling, he asked each student to type notes for one lecture during the semester. The materials for that class are beautiful and nice to read even years later. I hope that in the long term, more resources like this can be made and made available to the public, and I believe that students can/could/should play a role in this in the classroom.

OCW: How did you change the course materials or your instructional approach when teaching Introduction to Metric Spaces a second time?

Paige Bright: Mostly, the instructional approach was the same when teaching the course: highlight how analysis on metric spaces is similar to that on Euclidean space, discuss how it’s different with compact sets, and then a topic on metric spaces. The main thing that changed in 2023 was the topic.

In 2022, the topic for the course was differential equations– specifically talking about Picard iteration. While this topic is interesting in its own right, I decided to switch gears to talk about completions of metric spaces. But in either case, the topic was meant to apply to a different subject entirely: Picard iteration is an important skill in the study of differential equations, and completion of metric spaces is in the background of a lot of functional analysis. I hope in years to come, I (or someone else running the course) get to teach about similarly difficult topics that appear in analysis, such as the Stone-Weierstrass theorem, the Arzela-Ascoli theorem, the Weierstrass approximation theorem, the Baire Category theorem, and much much more.

OCW: What would you like to share about teaching 18.S190 that we haven’t yet addressed?

Paige Bright: In math, there are a great number of grey areas between classes; metric spaces are a great example of this. One of my biggest motivators for creating this course was the fact that this very important subject sometimes falls through the cracks in an undergraduate’s curriculum. This can make it harder to keep learning math in analysis and beyond. I hope we can keep identifying and creating resources for academic grey areas like this.

Also, I would like to thank the many students who helped provide feedback and guidance developing material and problem sets, including but not limited to Jacob Lerma, Omar Abdelghani, and Yuqiu Fu. I would also like to thank Professor Guth for helping develop the course and contributing interesting problems for problem sets and Professor Minicozzi for helping me navigate the logistics of creating this class.

Curriculum Information


Students were expected to have some experience with real analysis, such as might be gained from taking one of the various versions of 18.100 Real Analysis.

Requirements Satisfied



Various versions of 18.S190 Special Subject in Mathematics, each on a different topic, are offered each year during the Independent Activities Period (IAP).


The course was taught Pass / No Record, with grading based both on completion of assignments (three problem sets) and on participation.

Student Information


20 students

Breakdown by Year

Various, ranging from first-year undergraduate students to graduate students.

Typical Student Background

Students were expected to have some prior experience with real analysis. There were students who had just finished one of the versions of 18.100 Real Analysis, as well as those who simply wanted to know more about the subject. The majority of the students were enrolled as listeners.

How Student Time Was Spent

During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:

In Class (3 hours)

Met 2 times per week for 1.5 hours per session; 6 sessions total; mandatory attendance

Out of Class (9 hours)

Outside of class, students completed three problem sets.

Course Info

As Taught In
January IAP 2023
Learning Resource Types
Lecture Notes
Lecture Videos
Online Textbook
Problem Sets
Instructor Insights