Is there a word for the classification of a set as continuous or discrete? For example, in computer science, there can be zero, one, two, etc. parameters to a computer program, and this is called its "arity". Sets can be countable or uncountable. Is there some word I can use to say "this set's _" is continuous/discrete, or "this set has a continuous/discrete __". For example, although this sounds terrible, "this set's continuity is discrete" or "this set has a discrete continuity".  Perhaps granularity, coarseness, atomicity, separation, continuousness (biased word), discreteness (biased word)?
More possibilities:
Cohesion?  Cohesiveness?  Coherence?
 A: If you are talking abut more than the distinction between finite and infinite, you are probably talking about "topologically discrete."
A simple example is the rationals, which have a standard "non-discreet" topology, and the integers, which have a standard "discreet" topology. The two sets have the same cardinality, but the notions of continuity from the two sets are vastly different.
For another example, while $\mathbb N^\mathbb N$ is uncountable, the most obvious "topological" view of the set is as discrete topology. You can define other topologies on it, of course, but the simplest is the discrete one.
Basically, in topology, we are trying to define what functions from the set to another topology are "continuous." Under the so-called "discrete" topology, all functions are continuous, so you are considering all function from $X$. If $X$ does not have the discrete topology, then which functions from $X$ to $Y$ are continuous is determined by the topologies on both $X$ and $Y$.
A: In computer science a set $S$ is discrete if it is just a (finite or infinite) set, like a set of people, a set of colors used in coloring a graph, the sets ${\mathbb N}$ or ${\mathbb Z}^d$ or suitable parts of them. Variables taking values in such sets are called discrete variables. In dynamics we are talking about discrete time when the system under consideration is only observed at times $t\in{\mathbb Z}$.
A set $M$ or a variable taking values in this set is called continuous when $M$ is "built up" using real numbers in an essential way, e.g., when $M$ is a sphere $S^{n-1}\subset{\mathbb R}^n$, is the $n$-dimensional phase space of planetary motion, is a model of "color space", and so on. In these cases we need floating-point numbers to address individual points of $M$.
For purely mathematical purposes one can can characterize "discreteness" vs. "continuousness" in a succinct, but not very intuitive, way: A space $X$ is discrete iff for every $x\in X$ the set $\{x\}$ is a neighborhood of $x$. It follows that in a discrete space there are no interesting sequences $(x_n)_{n\geq1}$ converging to $x$.
