Quotient Spaces and Quotient Maps

Topology · Axiom Academy

Understanding how topology lets us "glue" points together Step 1: The Motivation - Gluing Points Together In topology, we often want to study spaces that arise from identifying or "gluing" certain points together. This is a fundamental construction that creates new topological spaces from old ones. The key idea: we want a systematic way to "collapse" certain subsets of a space to single points, or identify different points as being "the same" in our new space. Step 2: Setting Up - Equivalence Relations To make this rigorous, we use an equivalence relation. Given a topological space , an equivalence relation ~ partitions into disjoint equivalence classes. A relation ~ on a set is an equivalence relation if it satisfies: Given an equivalence relation ~ on , we form the quotient set , which is the set of all equivalence classes. The natural map defined by (sending each point to its equivalence class) is called the quotient map . The quotient topology on is defined by: In other words, a set is open in the quotient space if and only if its preimage is open in . Step 4: Example - The Circle from an Interval Consider with the equivalence relation: if . The quotient space is homeomorphic to the circle ! Watch as we glue the endpoints of the interval together to form a circle: Step 5: Example - The Torus from a Square Consider the square with the equivalence relation: if (glue left and right edges) if (glue top and bottom edges) The quotient space is homeomorphic to the torus !

This is the written version of the interactive lesson above. See the full Topology course.