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According to Introduction to Topology: Pure and Applied by Colin Adams,

Let $X$ be a topological space and A be a set (that is not necessarily a subset of $X$). Let $p: X \rightarrow A$ be a surjective map. Define a subset $U$ of $A$ to be open in $A$ if and only if $p^{-1}(U)$ is open in $X$. The resultant collection of open sets in $A$ is called the quotient topology induced by $p$, and the function $p$ is called a quotient map. The topological space $A$ is called a quotient space. (89).

As an example, I'd like to consider $\mathbb{ℝ}$ with the standard topology, and define

$$ p:\mathbb{R} \rightarrow \{a,b,c\} \ by \ p(x) = \left\{ \begin{array}{1l} c & \quad x < -10\\ b & \quad -10\leq x\leq10 \\ a & \quad x > 10 \end{array} \right. $$ Open sets in the quotient topology include $\{a\}$, $\{c\}$, $\{a,c\}$,$\{a,b,c\}$.

$\{b\}$ is not open since its preimage is not open in $\mathbb{R}$.

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    $\begingroup$ Is there a question here? $\endgroup$ – user332239 Dec 6 '16 at 21:01
  • $\begingroup$ Trying to make sure my example I'm offering is correct, i.e. the open sets, and not open example. $\endgroup$ – Learner Dec 6 '16 at 21:06
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    $\begingroup$ Yes, you’ve listed all of the non-empty open sets. $\endgroup$ – Brian M. Scott Dec 6 '16 at 21:08

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