Hereditary Good Properties and Net Weight

We will discuss what Ken Kunen called Hereditarily Good (HG) properties, and review super varieties of HG and the related Hereditarily Separable (HS) and Hereditarily Lindel\"of (HL) properties.

A regular Hausdorff  space X has the property HG (also called the pointed ccc) iff X has no weakly separated omega_1-sequences. Then X is super HG (suHG) iff for all assignments U = < (x_alpha, U_alpha) : \alpha < omega_1 > for X, where each x_alpha is in U_alpha and U_alpha is open, there is I in [omega_1]^{aleph_1} so that for all alpha,beta in I we have x_alpha in U_beta. Reducing the size of the subset I of omega_1 from aleph_1 to 2 elements weakens this to HG. Every space having countable net weight is trivially suHG.

We review the four distinct levels of HG properties and examples that separate levels, such as our MA(aleph_1) suHG space of uncountable net weight, and a few questions that remain, including whether there is a ZFC suHG space of uncountable net weight.