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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?

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    $\begingroup$ How is being continuous defined for sets? You simply mean not discrete? (And which definition of discrete do you use?) $\endgroup$
    – Tobias Kildetoft
    Commented Aug 6, 2013 at 18:30
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    $\begingroup$ Do you intend "continuous/discrete" to be a complete classification (every "set" is either one or the other but not both)? If so, do you think the set $[0,1]\cup\mathbb{Z}$ is a continuous or discrete subset of $\mathbb{R}$ with its usual topology? $\endgroup$
    – Zev Chonoles
    Commented Aug 6, 2013 at 18:32
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    $\begingroup$ @Doug: By the way, you seem to be unaware of the notion of "topological space" (Wikipedia), which I think is the natural setting of your question (also, sets need not be sets of real numbers). A continuum (Wikipedia) likely matches your intuitive idea of "continuous", and a discrete space (Wikipedia) likely matches your intuitive idea of "discrete", but there are many topological spaces that are not a continuum and are not discrete. $\endgroup$
    – Zev Chonoles
    Commented Aug 6, 2013 at 18:38
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    $\begingroup$ That does not sound like a useful distinction. What about intervals in the rationals? Why should all sets be considered as subsets of the reals? $\endgroup$
    – Tobias Kildetoft
    Commented Aug 6, 2013 at 18:38
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    $\begingroup$ Mostly, it seems like you'd want the word "topology." But it depends. There is nothing "topologically discrete" about an uncountable set, and there are non-discreet topologies on finite sets... $\endgroup$
    – Thomas Andrews
    Commented Aug 6, 2013 at 18:50

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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 $Y$.

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  • $\begingroup$ I would think that the indiscreet topology would be the simplest, since it only has two open sets. Although the discrete topology is a close second. $\endgroup$
    – Baby Dragon
    Commented Aug 6, 2013 at 19:00
  • $\begingroup$ Well, I'm assuming given the natural discrete topology on $\mathbb N$. The indiscreet topology is very rare indeed, since it allows for very few functions from it (but allows all functions to it.) $\endgroup$
    – Thomas Andrews
    Commented Aug 6, 2013 at 19:04
  • $\begingroup$ But I consider any non-$T_0$ topology is inherently "not natural," because intuition for topology has some separation. $\endgroup$
    – Thomas Andrews
    Commented Aug 6, 2013 at 19:06
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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 $x$.

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