Identification spaces:

Suppose that $X$ is a topological space and $\sim$ is an equivalence relation on $X$. The set of all equivalence classes is denoted by $X/\sim$ and $\eta : X \longrightarrow X/\sim$ denotes the projection map

$$\eta(x) = \overline{x}, \quad x \in X,$$

where $\overline{x}$ denotes the equivalence class of $x$. The space $X/\sim$ with the quotient topology induced by $\eta$ is called the identification space given by the equivalence relation. An important special case deserves mention as it is of frequent occurrence. Suppose that $A$ is a subset of a topological space then we consider the equivalence relation for which all the points of $A$ form one equivalence class and the equivalence class of any $x \in X - A$ is a singleton. That is to say all the points of $A$ are identified together as one point and no other identification is made. We shall refer to the resulting quotient space as the space obtained from $X$ by collapsing $A$ to a singleton.