Arithmetic Topology is motivated by an heuristic known as the "Knots and Primes" analogy. It was first noticed back in the 60s that certain objects in Number Theory seem to have similar properties to certain objects in topology. In particular, a number ring (e.g. the integers) is "like" a 3-manifold, and a prime of this number ring is "like" the embedding of a knot into this 3-manifold.

Of course naively this sounds ridiculous, how are the integers supposed to be like a manifold in any way? Essentially Mazur noticed that if you equip these arithmetic objects with something known as the étale topology, then the cohomologies of the arithmetic and topological objects look very similar and they obey very similar looking theorems.

An important note is that this analogy is purely a heuristic, so there is no mathematical proof that these objects are related. Arithmetic Topology uses this analogy to motivate potential theorems, results, or constructions. e.g. “There is some theorem Y in topology, what should the arithmetic analogue of this theorem look like?”. Ultimately in order to prove the arithmetic analogue one would still have to use number theory and not topology.