Question ii)
I don’t manage to prove that the intersection of open sets is still open.
Let’s assume that $U_1,U_2,...,U_m$ are in T. Let $f_0$ be in the intersection of $U_1,U_2, ...,U_m$. For all i in {1,...,m}, $f_0$ is in $U_i$.
We need to find a global $\epsilon$ and global $x_1,...,x_n$ for the particular set to be still in the intersection of $U_1,U_2, ...,U_m$, while we have such $\epsilon_i$ and $x_1i,...,x_ni$ only for each $U_i$...
One can maybe take $\epsilon$ = max($\epsilon_i$) but to find $x_1,...,x_n$ I really don’t know as they are not the same depending on i.
I don’t see neither how to prove Question iii).
Thank you very much
You want to prove that the intersection of two open sets is open. Let $f_0 \in U \cap V$. You want to find $\epsilon > 0, x_1, ... x_n \in [0,1]$ satisfying the condition for $f_0$.
You know that $f_0 \in U$ and $f_0 \in V$. Therefore, there exists $\epsilon_1 > 0 , x_{11}, ... x_{1m} \in [0,1]$ satisfying the condition for $f_0 \in U$, and a (possibly different) $\epsilon_2 > 0 , x_{21}, ... x_{2m} \in [0,1]$ satisfying the condition for $f_0 \in V$.
How can you combine them to get a collection that verifies $f_0 \in U \cap V$?
Hint: take $\epsilon < min(\epsilon_1,\epsilon_2)$ and the set $\{x_{11}, ... x_{1m}, x_{21}, ... x_{2m} \}$.