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Consider $\mathbb{R}$ with its standard topology. Suppose that $U$ is a subset of $\mathbb{R}$ consisting at most of proper intervals and isolated points.

Is there a standard terminology for $U$?

I would like a term to exclude all other types of subset, which are kind of pathological examples.

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    $\begingroup$ Do you consider (the singleton of) an isolated point an interval of the form $[a,a]$? $\endgroup$
    – Mark S.
    Aug 5 at 21:32
  • $\begingroup$ @MarkS. Nope. By "interval" I meant proper interval. I edited the question. $\endgroup$
    – MK7
    Aug 5 at 21:40
  • $\begingroup$ Standard term? Probably not, since just about any subset would look like that. $\endgroup$ Aug 5 at 21:49
  • $\begingroup$ @herbsteinberg Exactly. I thought there was a standard term to get rid of all "pathological" examples. $\endgroup$
    – MK7
    Aug 5 at 21:56
  • $\begingroup$ The entire real line consists of single integers and the open intervals between them so it doesn't seem very special. $\endgroup$
    – John Douma
    Aug 5 at 22:01

1 Answer 1

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If you are interested in finite unions of points and intervals, these are called the ”Semi-algebraic” sets in $\mathbb{R}$. The notion of semi-algebraic extends to $\mathbb{R}^n$ by considering all sets that can be picked out by a finite conjunction of statements of the form $p(x) = 0$ and $q(x) \geq 0$, where $p$ and $q$ are polynomials in $n$ variables with real coefficients. There is an interesting theorem called the “Tarski Seidenberg Theorem” that these sets are closed under coordinate projections.

The notion of semi-algebraic set corresponds with the notion of “definable with parameters” from mathematical logic when $\mathbb{R}$ is viewed as a real closed field. The Tarksi Seidenberg Theorem, mentioned above, exactly corresponds to the fact that the theory of real closed fields admits quantifier elimination.

If you’re interested in infinite unions, then you could say that these are the quantifier-free $\mathcal{L}_{\omega_1, \omega}$ definable sets with parameters, but I think that’s more of a mouth full than you’re looking for. This is not correct because I misunderstood the question.

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    $\begingroup$ Your last paragraph is incorrect: for example, $\mathbb{Q}$ is quantifier-freely-$\mathcal{L}_{\omega_1,\omega}$ definable with parameters. $\endgroup$ Aug 6 at 1:48
  • $\begingroup$ @NoahSchweber Maybe I’m miss understanding MK’s question, but isn’t $\mathbb{Q}$ a countable union of singletons? $\endgroup$
    – Joe
    Aug 6 at 1:51
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    $\begingroup$ Isolated points, not singletons. $\mathbb{Q}$ has no isolated points. $\endgroup$ Aug 6 at 2:03
  • $\begingroup$ Ahh you are correct. I misunderstood the question. $\endgroup$
    – Joe
    Aug 6 at 2:42

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