There are some common meanings to 'dense' in Mathematics.
- In Topology, a subset $S\subseteq X$ of a topological space $(X, \tau)$ is dense if the intersection of every non-empty open set with $S$ is non-empty.
- In linear order theory, if $(X,<)$ is linearly ordered and $S\subseteq X$, $S$ is dense if for every $x<y$ there is an $s\in S$ such that $x<s<y$.
In forcing or Martin's axiom, dense is defined as a subset $S\subseteq P$ of a partially ordered set $(P,\le)$ such that for all $x\in P$ there is an $s\in S$ such that $s\le x$.
The definition resembles more the definition of a filter-base (with the entire $P$ being the filter). Why is the term 'dense' used for this property? Is there a historical basis for this that predates forcing?
The definition can be made a particular case of the topological one by taking as basis for a topology on $P$ sets of the form $\{y\colon y\le x\}$ for $x\in P$. This seems contrived. The definition of dense on a linear order can also be made topological by taking the topology with open intervals as a base. This is more historical, as the topology on $\Bbb{R}$ was always defined via the linear order.
The relevant space isn't $P$ itself, but rather the space $\mathsf{MF}(P)$ of maximal filters through $P$ with topology generated by sets of the form $$U_p:=\{H: p\in H\}$$ for $p\in P$. For example, if we take $P$ to be $2^{<\omega}$ (an inessential variant of Cohen forcing), then $\mathsf{MF}(P)$ is just Cantor space $2^\omega$ with the usual topology. A set of conditions $D\subseteq P$ is dense in the forcing sense iff the corresponding set $$\bigcup_{p\in D}U_p$$ is dense - indeed, dense open - in the topological sense. And this is not contrived at all; this is the translation which makes clear the relationship between forcing and the Baire category theorem (see also this answer of mine).