About the Course


Topology is the mathematical study of shapes and space that considers questions such as, “What objects that visually seem quite different share the same properties?” One of the major fields of mathematics, topology possesses wide-ranging applications and beautiful theorems with far-reaching consequences. This course will introduce you and your classmates to point-set topology as you delve into bizarre notions of "space" and develop skills with rigorous, proof-based mathematics. You’ll begin by tackling the core concepts of sets, topologies, and continuous mappings before moving on to topological invariants such as compactness, connectedness, and the separation axioms. With these tools in hand, you will explore how to deform shapes and spaces without altering their fundamental properties. This knowledge allows you to see why it took 100 years for mathematicians to prove Poincaré’s 1904 conjecture about the nature of a sphere. Finally, you’ll survey different applications of topology, such as how the study of knots influenced our understanding of proteins, or how the study of manifolds led scientists to a deeper understanding of the topological shape of the universe.

Typical Class Size: 16-18

Course Overview

Testing and Prerequisites

  Math Verbal
Required Level CTY-Level Not required
Check your eligibility using existing test scores If you do not have existing test scores:

Students must achieve qualifying scores on an advanced assessment to be eligible for CTY programs. If you don’t have qualifying scores, you have several different testing options. We’ll help you find the right option for your situation.

Sign up for Testing Learn More

Course Prerequisites

Topology requires:

1 prerequisite

Geometry and Algebra II

Cost and Financial Aid

  • Tuition
    • Varies
  • Application fee
    • Nonrefundable Application Fee - $50 (Waived for financial aid applicants)
    • Nonrefundable International Fee - $250 (outside US only)

Financial Aid

We have concluded our financial aid application review process for 2024 On-Campus Programs. We encourage those who may need assistance in the future to apply for aid as early as possible.

Learn More

Course Materials

Students should bring basic school supplies like pens, notebooks, and folders to their summer program. You will be notified of any additional items needed before the course begins. All other materials will be provided by CTY.

Sample Reading

These titles have been featured in past sessions of the course, and may be included this summer. CTY provides students with all texts; no purchase is required.

  • Introduction to Topology, by Bert Mendelson

Technical Requirements

Students must bring a tablet, laptop computer, or Chromebook for use during the session. A smartphone will not be sufficient.

About Mathematics at CTY

Explore the study of shapes

Many of our courses allow students to describe the world around them in basic and profound ways. Our younger students learn about shape, scale, and proportion in Geometry and Spatial Sense. Middle School students explore beautiful real-world applications of lines; analyze data based on curves that fit a uniform, symmetric and bell-shaped, or skewed pattern in Data and Chance. And advanced students explore the underlying mathematics and fundamental characteristics of shapes, distance, and continuous deformations in our proof-based Topology course.

Dive deep into logic and reasoning

Our courses in formal logic give you the tools to question the world around you. Inductive and Deductive Reasoning introduces younger students to different types of reasoning, as well as the strengths and weaknesses inherent in various forms of critical analysis. Older students explore how logical reasoning can explain (or fail to explain) counter-intuitive results in Paradoxes and Infinities, or take a more rigorous approach to formal logic in Mathematical Logic.

Meet our instructors and staff