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*.

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 S^{1}.

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 S^{2}.

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 *f*o*p* 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* = *h*o*p*.

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.

- 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.