Aims and Outcomes for “A Gentle Introduction to Topology”

A course I would like to teach would be a gentle introduction to topology, in which I could help enterprising students bridge the gap between calculus and higher mathematics. The course would introduce concepts in the foundations of mathematics and move into a discussion of the field of topology. Ultimately I would like to lead them back to more familiar examples and  let the students analyze them with their new-found tools.

AIMS

• The main themes or ideas I will emphasize are the notions of a topology on a set, the structures of topological spaces, the properties of topological spaces, and an introduction to the idea of a manifold.
• The big picture or storyline is that topological structures exist everywhere in mathematics and that a basic understanding of topology can dramatically improve one’s intuition with respect to other areas of mathematics.
• The main questions I am interested in are how to define a topology on a set, what properties certain topological spaces have, how does this relate to the students’ experiences in calculus.
• The mental model I am proposing is one of intellectual curiosity and pleasure in math for math’s sake.

OUTCOMES

• To answer these questions, I want my students to become more skillful in determining if a given space is a topology, analyzing a topological space to determine its properties, and honing their skills at doing proofs.
• The mental model that my students may bring with them and that I want to challenge is that mathematics is about numbers and is only useful if it can be applied.

STYLE

• The diction and tone that I want to use are casual and conversational. I want my students to be able to engage me as a peer; another traveller on the road of life who happens to share a love and passion for mathematics.