Quotient topology

Definition

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 $A$.