# The possible spectra of compactness

**Definition.** *Given a property of mathematical structures of a given class, we say that $\kappa$ is a compact cardinal for the given property, if every structure of our class has the property, given that every ``smaller'' substructure has the property. $\kappa$ is a weakly compact cardinal for the property if the above holds for structures of ``size'' $\le \kappa$.*

*In the above, ``smaller'' and ``size'' may have different meanings for different properties. The (weak) compactness spectrum of the property is the class of cardinals that are (weakly) compact for the given property.*

Since any compactness is related to large cardinals properties, the compactness spectra depends very much on the universe of set theory. So the question is really: ``which cardinals can consistently be compact for the given property''?

For many properties it will be interesting the analyse the possible compactness spectra for these properties. Typical examples are:

A group being free.

An abelian group being free.

A collection of countable sets having a transversal (a one-to-one choice function).

A collection of countable sets that can be disjointing. (Namely that one can remove a finite subset from each of the members of the family such that the resulting family will be mutually disjoint.)

A topological space being collectionwise Hausdorff.

A compact space being Corson. (Namely being homeomorphic to a subspace $X \subseteq [0,1]^\kappa$ such that for all $x \in X$ $\{\xi < \kappa | x(\xi) \ne 0\}$ is countable.)

In the talk we shall survey some of the known results about compactness spectra, as well as many open problems.