How do we call a subset of $\mathbb{R}$ made up of intervals and isolated points? 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.
 A: 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.
