Let $\{f_i\}$ be the set of all bounded continuous real functions with respect to the usual topology on $R$, and $\{g_i\}$ be the set of all continuous real functions with respect to the usual topology on $R$. Moreover, let $T_1$ be the weak topology on $R$ generated by $\{f_i\}$ and $T_2$ be the weak topology on $R$ generated by $\{g_i\}$. What are the differences between $T_1$ and $T_2$? Are they the same?
Many thanks in advance.
Both $T_1$ and $T_2$ coincide with the usual topology on $\mathbb{R}$. Indeed, if $(a,b)$ is a bounded open interval then considering the bounded continuous function $$f(x)=\begin{cases} & 0 & x <a \\ &x-a& x \in [a,b] \\ & b-a & x >b \end{cases}$$ we conclude that $(a,b)=f^{-1}((0,b-a)) \in T_1$. Since each open set is union of such intervals, $T_1$ (and so $T_2$, which is bigger) is the whole euclidean topology.