Chain Conditions of Relations.

• Compact – indeed sigma-compact – topological groups have the countable chain condition (ccc).’
• Function spaces with a weak topology – like – are ccc, indeed property K.
• Forcing with a ccc partially ordered set preserves cardinals.

‎‎‎These are three, from many, examples where chain conditions (countable chain condition, Property K, calibre etc) of spaces or partially ordered sets play a key role in topology and set theory.

‎‎‎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. A central role is played by the Tukey order on relations, which enables us to transfer information about chain conditions from one relation to another.

‎‎‎In particular we look at relations connected to bases and networks of spaces.