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Let $P$ be the prime numbers in $\Bbb{Z}^+$. Define a topology on $\Bbb{Z}^+$ by taking as basis sets of the form $P\cdot P \cdot \ldots \cdot P \ (k \text{ times})$, where the $\cdot$ is semigroup multiplication between each two pair of elements, $k \geq 0$. Then $P^k \cap P^l = \varnothing$ vacuosly. What's it's name and any links?

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This is not really a topology so much as a partition. Since $\mathbb{Z}^+ = \coprod_{k\geq 0} P^k$, each $P^k$ is both open and closed in this topology, and so all we've really done is broken up $\mathbb{Z}^+$ into the sets $P^0, P^1, P^2,\ldots$ and given each one the trivial topology.

Many mathematicians have divided up the natural numbers by number of prime divisors with multiplicity, but I don't think there's a special name for it.

Also, we haven't used the additive structure of $\mathbb{Z}$ at all. As such, we can think of $\mathbb{Z}^+$ as the free abelian monoid on countably many generators, with isomorphism $\oplus_{i=0}^\infty \mathbb{N}\to \mathbb{Z}^+$ given by $(a_0,a_1,\ldots)\mapsto \prod_i p_i^{a_i}$, where $p_0, p_1,\ldots$ are the primes.

For this reason, anything we do with $\mathbb{Z}^+$ that doesn't make use of addition will not have much number-theoretic significance, since it will all boil down to studying a free monoid.

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