Cardinal characteristics of the continuum (CCC) are cardinals which are associated with naturally arising combinatorial properties of "the continuum".

The reason that "the continuum" is qualified, is that sometimes it is better to think about it as the Cantor space, other times as the Baire space, and sometimes as $\Bbb R$ itself, or as $[0,1]$. While different, they are all Polish spaces, and therefore Borel isomorphic.

For example, we can ask what is the smallest size family of functions $\cal F\subseteq\omega^\omega$ such that for every $g\in\omega^\omega$ there is some $f\in\cal F$ such that $g\leq f$ everywhere, or at least $g\leq^f$, namely there is some $m$, such that for all $n>m$, $g(n)\leq f(n)$.

One can show that this cardinal, also known as *the dominating number* and usually denoted by $\frak d$, is uncountable, and of course bounded by $2^{\aleph_0}$. We can prove, for example, that its cofinality is uncountable. And it is consistently taking many different values.

Some are more topological or measure-theoretic in nature, e.g. what is the smallest cardinality of a non-meager set, or how many null sets are needed to cover the whole space.

There are many more CCCs, some more famous (e.g. $\frak p$ and $\frak t$ which made headlines over the last summer), and some are less famous (e.g. $\frak h$ the shattering number).

Where can I find a brief history of the research into CCCs?

Set theoretic real analysis. $\endgroup$ – Dave L Renfro Dec 14 '17 at 14:32