Weak duals and neighbourhood assignments (Q1725673)

For a topological property $\mathcal{P}$ a space $(X,\mathcal{T})$ is weakly dually $\mathcal{P}$ provided for each neighbourhood assignment $\nu:X\to \mathcal{T}$ there is a $\mathcal{P}$ subspace $Y\subset X$ such that $\bigcup\{\nu(y)\mid y\in Y\}$ is dense in $X$. The property of having the discrete countable chain condition is weakly self-dual on weakly regular spaces while the properties weakly Lindelöf, separable and of having the countable chain condition are all weakly self-dual on Baire developable spaces. Related conditions bounding the cardinality of a space by the continuum are presented; for example a Baire, Hausdorff space that has the weakly (dually) discrete countable chain condition and a rank 2 (3) diagonal.