TITLE: Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ that satisfies these conditions?
QUESTION [6 upvotes]: Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ from the reals into the power set of the naturals such that

for any $x\in\mathbb{R}$ the set $f(x)$ is infinite, and
for any distinct $x,y\in\mathbb{R}$ the intersection $f(x)\cap f(y)$ is finite?

REPLY [3 votes]: First, get an injection $g : \mathbb{R} \to P(\mathbb{N} \times \mathbb{N})$ satisfying these conditions by setting $g(x) := \{ (n, \lfloor 10^n x \rfloor) : n \in \mathbb{N} \}$.  Now, using the fact that $\mathbb{N} \times \mathbb{N}$ is countable, can you see a way to convert $g$ to a function $f$ satisfying the conditions?