I am reading something on completion of metric spaces and it says:
Let $\hat S$ be $\mathcal{C}$ modulo equivalence relationship of co-Cauchy sequences. Where $\mathcal{C}$ is the set is all Cauchy sequences on $S$.
I think I have encountered this in other contexts say in differential geometry, you see people drawing $\square \text{\~}$, defined as $\square$ modulo equivalence relationship. It seems to me this is pretty hand wavy but I don't understand what modulo mean in this context.
I tried looking up this on google but unfortunately it gave me a bunch of stuff on number theory about modulo operator.
This means that $\hat{S}$ is the set of all equivalence classes of $\mathcal{C}$. It assumes that we have in mind some specific equivalence relation $\sim$ on $\mathcal{C}$.
In other words, every element $s$ of $\hat{S}$ is a nonempty subset of $\mathcal{C}$. The following two conditions hold:
If $x \in s$ then we have $y \in s$ iff $x \sim y$
For every $x \in \mathcal{C}$ there exists a unique $s \in \hat{S}$ with $x \in s$.