Exercise 9, § 2

a) With the hypotheses of Proposition 4 of no. 3, show that the coarsest topology on $X$ for which the $f_i$ are continuous is also the finest topology $\mathfrak{T}$ on $X$ which has the following property: every mapping $g$ of a topological space $Z$ into $X$ such that the $f_i \circ g$ are continuous is continuous (with respect to $\mathfrak{G}$).

b) With the hypotheses of Proposition 6 of no. 4, show that the finest topology on $X$ for which the $f_i$ are continuous is also the coarsest topology $\mathfrak{T}$ on $X$ which has the following property: every mapping $g$ of $X$ into a topological space $Z$ such that the $g \circ f_i$ are continuous is continuous (with respect to $\mathfrak{T}$).