Constructing the Circle as a Quotient

Topology · Axiom Academy

EXAMPLE Quotient Spaces and Identification Building topological spaces by gluing points together Problem 1: The Circle from an Interval We start with the unit interval and define the equivalence relation where: This means we identify (glue together) the endpoints 0 and 1, while all other points are only equivalent to themselves. The quotient space is , which consists of equivalence classes: The equivalence class contains both endpoints For any , the class contains only that point A set in the quotient space is open if and only if its preimage under the quotient map is open in . Problem 2: The Torus from a Square These identifications must be compatible at corners: all four corners are identified to a single point. The construction proceeds in stages: First, glue the left and right edges: this creates a cylinder Then, glue the top and bottom circles of the cylinder together: this creates the torus The quotient map is defined by , where we use coordinates on the torus as a product of circles. Problem 3: The Möbius Strip with a Twist Starting with , we identify the vertical edges with a twist : Notice the twist: the point is identified with , and with . Non-orientable: If you travel around the strip following a normal vector, you return to the starting point with the normal reversed One boundary component: The top and bottom edges of the square are NOT identified, forming a single boundary circle One-sided surface: Unlike a cylinder, the Möbius strip has only one side

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