From Rectangles to Tori: The Quotient Topology

Publication Date

2026

Presentation Length

30 minutes

College

College of Sciences & Mathematics

Department

Math and Computer Science, Department of

Student Level

Undergraduate

Faculty Mentor

Adam Cartisano

Presentation Type

Talk/Oral

Summary

The quotient topology provides a way to construct new topological spaces by “gluing” or identifying points in an existing space. This presentation defines quotient maps, saturated sets, and the induced quotient topology on partitions (identification spaces). The goal is to build on understanding how complex spaces are derived by “gluing” simple shapes together. Examples include collapsing boundaries to form spheres and gluing edges of rectangles to obtain tori. Key results cover when restrictions of quotient maps remain quotient maps, the universal property for continuous functions out of quotient spaces, and induced homeomorphisms. Applications highlight geometric constructions and subtle behaviors (e.g., products and Hausdorff conditions).

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