What is the meaning of $ \mathbb{R} \bmod T$?

Question

What is the meaning of $ \mathbb{R}\bmod\ T$ for a fixed $T>0$. The set of all equivalence classes?

And how can I derive a function of the form $f:\mathbb{R}^d \to \mathbb{R}\bmod T$?

Answer 1

I would assume

$$ x \equiv y \mod T \Leftrightarrow \exists n \in \mathbb Z: x- y = nT. $$

This then yields a quotient group (namely, the one of $(\mathbb R, +)$ divided by the subgroup $\{nT|n \in \mathbb Z\}$) with well-defined addition.

Your function would then come from the composition $\mathbb R^d \to \mathbb R$ (eg. the projection) with the quotient map $\mathbb R \to \mathbb R / \langle T \rangle$.