Actually I have to ask for the quotient topologies in 3 different situations (which come from Ex. 5.5.7 of Intro to Topology by Min Yan).
Let's consider a function $f(x): [a, b] \to [\alpha = \underset{[a, b]}{\min} f, \beta = \underset{[a, b]}{\max} f]$.
- Prove if $[a, b]$ has the usual topology, the quotient topology on $[\alpha, \beta]$ is the usual topology.
My attempt is:
$f$ is continuous from $[a, b]_{\text{usual}}$ to $[\alpha, \beta]_{\text{usual}}$. Then $[\alpha, \beta]_{\text{quotient}}$ is finer than $[\alpha, \beta]_{\text{usual}}$.
On the other hand, the identity map Id$(x) = x: [a, b]_{\text{usual}} \to [a, b]_{\text{usual}}$ is also continuous, so $f(x) = f \circ$ Id$(x): [a, b]_{\text{usual}} \to [\alpha, \beta]_{\text{quotient}}$ is also continuous. This implies the $[\alpha, \beta]_{\text{quotient}}$ is coarser than $[\alpha, \beta]_{\text{usual}}$. Therefore $[\alpha, \beta]_{\text{quotient}} = [\alpha, \beta]_{\text{usual}}$.
I want to ask if this is correct & vigorous enough.
- Prove if $f$ is increasing, and $[a, b]$ has the lower limit topology, the quotient topology on $[\alpha, \beta]$ is the lower limit topology.
I'm stuck at showing that if $f(x) = \beta$, there is a $[c, d)$ being a subset of an open subset $U$ of $[\alpha, \beta]$. Is there any way to do so? Or, is there another approach to prove the statement?
- Find a general sufficient condition on $f$ s.t. if $[a, b]$ has the lower limit topology, the quotient topology on $[\alpha, \beta]$ is the usual topology.
The lower limit topology is finer than the usual one, right? But I have no idea what kind of conditions is needed. I only know if the function is already continuous in the case of usual $\to$ usual, [finer than usual] $\to$ usual is still continuous.
Many thanks again.