Chain Conditions of Relations and Hereditarily Good Spaces

There are many examples where chain conditions (countable chain condition, Property K, calibre \omega_1 etc) of spaces or partially ordered sets play a key role in topology and set theory. Compact topological groups are ccc; function spaces with a weak topology - like C_p(X) - are ccc, indeed property K; and forcing with a ccc partially ordered set preserves cardinals.

In this talk we will extend the idea of `chain conditions’ to more general relations arising in topology. This will allow us to understand and organize what we currently know, and raises many new problems. In particular we look at relations connected to bases and networks of spaces. This will lead us to investigate the class of hereditarily good spaces, that have recently received considerable attention.