Theorem 4.4:

Let $X$ and $Y$ be topological spaces and $f : X \longrightarrow Y$ be a surjective continuous map such that the given topology on $Y$ agrees with the quotient topology on $Y$ induced by $f$. Define an equivalence relation $\sim$ on $X$ as follows:

$$x_1 \sim x_2$$    if and only if $$f(x_1) = f(x_2), \quad x_1, x_2 \in X$$

The identification space $X/\sim$ is homeomorphic to $Y$ via the map $\phi : X/\sim \longrightarrow Y$ given by

$$\phi(\overline{x}) = f(x).$$