On function spaces

For certain properties $\mathfrak{P}$ of topological $T_0$-spaces, we prove that an arbitrary $T_0$-space $\mathbb{Y}$ has property $\mathfrak{P}$ if and only if the function space $\mathbb{C}(\mathbb{X},\mathbb{Y})$ endowed with the pointwise convergence topology possesses $\mathfrak{P}$ for some (and therefore, for each) $[\alpha^\ast-]$space $\mathbb{X}$.