Quotient Space - Examples

Examples

• Gluing. Often, topologists talk of gluing points together. If X is a topological space and points are to be "glued", then what is meant is that we are to consider the quotient space obtained from the equivalence relation a ~ b if and only if a = b or a = x, b = y (or a = y, b = x). The two points are henceforth interpreted as one point. Such gluing is commonly referred to as the wedge sum.
• Consider the unit square I2 = × and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then I2/~ is homeomorphic to the unit sphere S2.
• Adjunction space. More generally, suppose X is a space and A is a subspace of X. One can identify all points in A to a single equivalence class and leave points outside of A equivalent only to themselves. The resulting quotient space is denoted X/A. The 2-sphere is then homeomorphic to the unit disc with its boundary identified to a single point: D2/∂D2.
• Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y if and only if x−y is an integer. Then the quotient space X/~ is homeomorphic to the unit circle S1 via the homeomorphism which sends the equivalence class of x to exp(2πix).
• A vast generalization of the previous example is the following: Suppose a topological group G acts continuously on a space X. One can form an equivalence relation on X by saying points are equivalent if and only if they lie in the same orbit. The quotient space under this relation is called the orbit space, denoted X/G. In the previous example G = Z acts on R by translation. The orbit space R/Z is homeomorphic to S1.

Warning: The notation R/Z is somewhat ambiguous. If Z is understood to be a group acting on R then the quotient is the circle. However, if Z is thought of as a subspace of R, then the quotient is an infinite bouquet of circles joined at a single point.