After revising my notes I have two problems still on topology, and as my midterms are near I need some desperate help. For problem 1, I think I can solve it, but for problem 2, I have asked two days ago but I still seem to be having a trouble. Can anyone however check if my problem 1 is done correctly?
$T_Y=\{V⊂Y:f^{−1}(V)∈T_X\}$ is the definition of the quotient topology on $Y$ given a function $f:X\rightarrow Y$, and $X$ is a topological space with the topology $T_X.$
Problem 1: For determining the quotient topology on the unit circle $\Psi(t)=e^{2\pi i t}:\mathbb{R}_l\rightarrow S^1$, where $S^1$ is the unit circle. This map is clearly onto, and the openness of a subset$ U \subset S^1$ in the quotient topology implies that for any $\Psi(t)\in U$, there will be some open ball $B(t-\delta,t+\delta)$ such that $\Psi(B)$ is an open arc on the unit circle. Such open arcs are topological basis of $S^1$ unit circle, as a topological subspace of $\mathbb{R}^2$ of open rectangles. Thus it is not hard to see that the quotient topology is the same as the usual subspace topology on the unit circle.
Problem 2: 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 think I get a little grasp on how to do quotient topology problems, however whenever I see like max min symbols, it really confuses me on how to treat the problems. I would appreciate some help with problem 2, as I really need to understand how this quotient topology works.