I have a question on quotient topology which I just do not understand the authors explanation at all for this example, and that is:
Example: 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 have been doing some examples on this topic quotient topology, but I can only find one relevant example in my notes which was determining the quotient topology on a function $f=x^2$ from the lower limit topology on $\mathbb{R}$ to $[0,\infty)$, this example I could understand, but although my concepts with open sets, and continuity are getting better, now it seems like I am stuck on trying to understand quotient topologies.
I would really appreciate some clear help to help me understand topology better.
Hint: To show two topologies are equivalent you need to show that one is a subset of another and vice versa, so that they both (as sets) are equal.