Notation in point set Topology

Question

In $\mathbb R \setminus \mathbb Q$ and $\mathbb R /\mathbb Q$, what do these ("$\setminus$","$/$") symbols between the sets of real and rational numbers mean?

Answer 1

Typically $\mathbb{R}\backslash\mathbb{Q}$ is the set theoretic difference, i.e. the set of all irrationals in this case.

While $\mathbb{R}/\mathbb{Q}$ is the quotient group. It can also mean the result of collapsing a topological subspace to a point. Or the orbit space of $\mathbb{Q}$ acting on $\mathbb{R}$ and potentially other things. So it depends on the context. Either way it is some form of a quotient set.

Edit: As noted by @AlephNull, it can also stand for a field extension. In which case it doesn't arise from an equivalence relation, making the notion even more context sensitive.

Answer 2

Depends on the context:

$\mathbb{R} \setminus \mathbb{Q}$ is the set difference between the reals and the rationals, so it equals the set of irrationals.

$\mathbb{R}/\mathbb{Q}$ can mean the quotient of the group of reals by its subgroup of the rationals, (which also gives a topological group) or it can denote a quotient space of the reals where we identify the subset of rationals to a single point.