Chain Conditions and Pixley-Roy/Ocan Hyperspaces

After a review of the history of chain conditions in topology, starting with Souslin's Problem, we will see van Douwen's solution - via a chain condition - to the problem of determining which spaces have a separable compactification. Then we turn to Pixley-Roy/Ocan hyperspaces, discuss their key properties, before showing how they give an example of a non-separable space with separable compactification. Finally we consider chain conditions on Pixley-Roy/Ocan hyperspaces, including the still open problem of when they have the countable chain condition.