Here is a topology question, and $T_{st}$ stands for standard topology:
Consider the function f : [−1, 1] → [0, 1] defined by
$$f(t)= \begin{cases} t& \text{0 $\leq$ t $\leq$ 1}\\ -t& \text{-1$\leq$ t $\leq$ 0} \end{cases}$$ (a) Is f : ([−1, 1],$T_{st}$) → ([0, 1],$T_{st}$), continuous?
(b) Is f : ([−1, 1],$T_{st}$) → ([0, 1],$T_{st}$) a homeomorphism?
(c) Is ([−1, 1],$T_{st}$) homeomorphic to ([0, 1],$T_{st}$)?
What I got so far:
(a) just use calculus from [0,1], $lim_{t -> 0+}$ t = 0, and from [-1,0] $lim_{t -> 0-}$ t = 0， since LHS = RHS, it's continuous. And since f(t) always goes to a value in the range [0,1], so f is continuous.
(b)it's bijective, and f is continuous, but I'm not sure how to check the inverse of this function.
(c) I think if (b) is homeomorphism, then c must be. But if (b) is not, then I'm not sure how to find the function that goes from ([−1, 1],$T_{st}$) to ([0, 1],$T_{st}$) maybe f(t) = |t|? which takes the absolute value.
Please point out whether I'm correct on each question, thanks in advance for any help!
$$f(t)= \begin{cases} t& \text{0 $\leq$ x $\leq$ 1}\\ -t& \text{-1$\leq$ x $\leq$ 0} \end{cases}$$
(a) Is $f : ([−1, 1],st) → ([0, 1],st)$, continuous?
Yes, it is continuous.
(b) Is $f : ([−1, 1],st) → ([0, 1],st)$ a homeomorphism?
No, it is not one-to-one so it is not homeomorphism.
(c) Is $([−1, 1],st)$ homeomorphic to $([0, 1],st)$?
Yes, the function $f(x)=(x+1)/2$ is a homeomorphism from $[-1,1]$ to $[0,1]$