Let $h$ be a (real-valued) continuous function on some closed interval $[c,d].$ Let, $\lambda =\min_{[c,d]}h$ and $\gamma = \max_{[c,d]}h$ . Also, $h$ can be considered as a surjective map (onto) from $[c,d] \rightarrow [\lambda,\gamma]$.

**Question** How can one show that if $[c,d]$ has the usual topology, then the quotient topology on $[\lambda,\gamma]$ is also the usual topology ?

I am genuinely frustrated because this is an example from the quotient topology chapter, but the solution to this the author gave I cannot understand at all. I would really appreciate some help.

Note the quotient topology I am working with here is given by: a quotient topology on $Y$ is defined to be $T_Y=\{ V\subset Y : f^{-1}(V) \in T_X\}$,where $f:X\rightarrow Y$, and a topological space $X$ with topology $T_X.$