Quotient topology

Quotient topology by an equivalence relation
Suppose $$X$$ is a topological space and $$\! \sim$$ is an equivalence relation on $$X$$. In other words, $$\! \sim$$ partitions $$X$$ into disjoint subsets, namely the equivalence classes under it. The quotient space of $$X$$ by $$\! \sim$$, or the quotient topology of $$X$$ by $$\! \sim$$, denoted $$\! X/\sim$$, is defined as follows:


 * As a set, it is the set of equivalence classes under $$\sim$$.
 * The topology on it is defined as the finest topology possible so that the quotient map $$\! q:X \to X/\sim$$, that sends every element $$x \in X$$ to its equivalence class, is a continuous map. Equivalently, the open sets in the topology on $$\! X/\sim$$ are those subsets of $$\! X/\sim$$ whose inverse image in $$X$$ (which is the union of all the corresponding equivalence classes) is an open subset of $$X$$.

The map $$q:X \to X/\sim$$ is a quotient map. In fact, the notion of quotient topology is equivalent to the notion of quotient map (somewhat similar to the first isomorphism theorem in group theory?)

Quotient topology by a subset
Suppose $$X$$ is a topological space and $$A$$ is a subset of $$X$$. The quotient space $$X/A$$ is defined as the quotient space $$\! X/\sim$$, where $$\sim$$ is the equivalence relation that identifies all points of $$A$$ with each other but not with any point outside $$A$$, and does not identify any distinct points outside $$A$$. In other words, all points of $$A$$ become one equivalence class, and each single point outside $$A$$ forms its own equivalence class.

Note that a notation of the form $$X/A$$ should be interpreted carefully. In case $$X$$ is a topological group and $$A$$ is a subgroup, this notation is to be intepreted as the coset space, and not in terms of the description given above. Context is extremely important.

Quotient topology by a subset with based topological space interpretation
Suppose $$X$$ is a topological space and $$A$$ is a subset of $$X$$. We may be interested in the pair of topological spaces $$(X,A)$$. In this context, $$X/A$$ (as defined above) is often viewed as a based topological space, with basepoint chosen as the equivalence class of $$A$$.