# subsets of $\mathbb{R}^+$ closed under addition

No one's answered this question so here's a simpler question: Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition? In particular, how about those that are topologically closed?

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You can take any closed set of positive reals, close it with respect to addition. After that you can include $0$ if you want to. The only remaining set will be $[0,\infty)$. – Anton Petrunin Apr 2 '12 at 22:31
In particular, if that closed set $C$ has a minimum positive element, then it generates a closed additive semigroup $S:= C\cup (C+C) \cup (C+C+C)\dots$. If $C$ has no minimum positive element, it generates a dense semigroup. – Pietro Majer Apr 2 '12 at 22:42
I don't understand either comment. Help? – Benjamin Steinberg Apr 2 '12 at 22:44
Anton’s comment gives a recipe how to construct all closed subsets of $\mathbb R^+$ closed under addition. – Emil Jeřábek Apr 2 '12 at 22:53
What does he mean by the only set remaining? Also does one not have to close again under the topology? – Benjamin Steinberg Apr 2 '12 at 22:58

Update Although individually subsemigroups of $\mathbb{Z}$ are "easy" - they consist of an arithmetic progression plus a finite "garbage", the subsemigroups of $\mathbb{Q}_+$ are already very complicated. Every cancelative countable commutative semigroup without torsion (and there are lots of those) embeds into $\mathbb{R}$. That is because it embeds into a commutative countable group which, in turn, embeds into a product of copies of $\mathbb{Q}$ which is a subgroup of $\mathbb{R}$.
@Ben: Michael Hardy wanted a catalog. That is impossible. Individually subsemigroups of $\mathbb{Z}_+$ are not too difficult to describe: they are virtually arithmetic progressions. Subsemigroups of $\mathbb{R}_+$ and even $\mathbb{Q}_+$ are much harder. An interesting question is to describe subsemigroups of $\mathbb{Q}_+$ that are residually finite. I spent quite some time on that when I was an undergraduate student - without much success. Some non-trivial problems from descriptive topology appeared, as far as I remember. – Mark Sapir Apr 2 '12 at 22:51