Quotient space

In topology, a quotient space is (intuitively speaking) the result of identifying or "gluing together" certain points of some other space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. The operation can be thought of (very informally indeed) as the act of "dividing" the input space by the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division.

Formally, suppose X is a topological space and ~ is an equivalence relation on X. We define a topology on the quotient set X/~ (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open if and only if their union is open in X.

Examples

Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y iff x-y is an integer. Then the quotient space X/~ (also written as R/Z) is homeomorphic to the unit circle S¹.

As another example, consider the unit square X = [0,1]×[0,1] 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 X/~ is homeomorphic to the unit sphere S².

Properties

Let p : X → X/~ be the projection map which sends each element of X to its equivalence class. The map p is continuous; in fact, the topology on X/~ is the finest (the one with the most open sets) which makes p continuous. The map p is in general not open.

If Y is some other topological space, then a function f : X/~ → Y is continuous if and only if fop is continuous.

If g : X → Y is a continuous map with the property that a~b implies g(a)=g(b), then there exists a unique continuous map h : X/~ → Y such that g = hop.

The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is constantly being used when studying quotient spaces.

Compatibility with other topological notions

• Separation
  • A quotient space of a Hausdorff space need not be Hausdorff.
• Connectedness
  • If a space is connected or path connected, then so are all its quotient spaces.
  • A quotient space of a simply connected or contractible space need not share those properties.
• Compactness
  • If a space is compact, then so are all its quotient spaces.
  • A quotient space of a locally compact space need not be locally compact.

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