The problems we shall discuss are examples of compactness (or dually a

reflection) problems. A typical compactness property is a statement that a

given structure has a certain property, provided smaller some structures have

this property. Reflection property is a dual statement, namely if a structure

has the property then there is a smaller substructure having the property. The

notion of "smaller substructure " may depend on the domain we talk about. Thus

for an algebraic structure "smaller substructure" typically means a subalgebra

having a smaller cardinality. For topological spaces "smaller substructure "

may mean a continuous image of the space of smaller weight.

Compactness problems tend to form clusters, which share the same pattern. For

instance sharing the same cardinals which are compact for the given property.

In this talk we shall survey some of these patterns. But we shall concentrate

on the problem of compactness for a compact space being Corson.

A compact space is a Corson compact if it can be embedded into $\Sigma\left(

R^{k}\right) $ where $\Sigma\left( R^{k}\right) $ is the the subspace of

$R^{k}$ (with the product topology) of those sequences which are non zero only

on countably many coordinates. The compactness problem for Corson compacta is

whether a space is Corson compact if compact given that all its continuous

images of small weight are Corson. We shall report on some ongoing work about

this problem.