Introduction to Metric Spaces and Topology

Project Description

This was a project for 1st and 2nd year undergraduate students and aimed to give a brief introduction to Topology. The project first built on the foundations of the concept of sequences and convergence taught in 1st year Mathematics courses but then moved on quickly to defining metric spaces. Having studied metric spaces the project then introduced the concept of topological spaces and some of their associated properties.
The project was mentored by Kushlam Srivastava, a final year BS-MS dual degree student in the Department of Mathematics and Statistics, IIT Kanpur. The following 1st year undergraduate students participated in and completed the project - Aaryan Srivastava, Kalash Talati, Neha Pattanaik, Ishita Agarwal, Srujan Killada, Suyash Kumar Patha and Vedant Nimbhorkar.
The participants worked on a combined project report which can be found here.

Mentors

  • Kushlam Srivastava

Learning Roadmap

  1. Firstly, the concepts of sequences, the convergence of sequences and their various criteria along with limits of functions and their continuity were revised.
  2. Then the definition of a metric was introduced along with some examples.
  3. This was followed by open and closed sets along with some examples and equivalent definitions.
  4. We then defined the continuity of a function on metric spaces both using the metric itself and then using only open sets.
  5. Finally, topological spaces were introduced along with the definition of continuity on topological spaces and homeomorphisms.
  6. At the end the concepts of compactness and connectedness in topological spaces were introduced.
  7. This was followed by presentations on some specific topics by each of the participants.
    Their presentation slides can be found here and their final project reports can be found here.

Resources

  1. Introduction to Topology and Modern Analysis by G F Simmons.
  2. M.A. Armstrong - Basic Topology.