A spatial microscopic configuration of a set of $N$ non-interacting point-like particles in 3D is given once we have all the positions, i.e. the set of all the position vectors $Q=\{{\bf r}_1, \dots , {\bf r}_N \}$.

If the particles are confined in a finite volume, each cartesian component of the position ${\bf r}_i$ is an element of a subset of the real numbers $R$, each position vector ${\bf r}_i$ is inside a finite volume $V$, and the $3N$ position $Q$ is an element of a finite volume subset of $R^{3N}$, the $3N$-fold cartesian product of $R$, of volume $V^{3N}$.

Notice that the continuity of the values of the positions has nothing to do with the spatial dimensionality. When we say that the continuous values of the position ${\bf r}_i$ belong to a 3D space, we mean that we need $3$ real numbers to uniquely assign a position in the space.

A spatial microscopic configuration of a set of $N$ non-interacting point-like particles in 3D is given once we have all the positions, i.e. the set of all the position vectors $Q=\{{\bf r}_1, \dots , {\bf r}_N \}$.

If the particles are confined in a finite volume, each cartesian component of the position ${\bf r}_i$ is an element of a subset of the real numbers $R$, each position vector ${\bf r}_i$ is inside a finite volume $V$, and the $3N$ position $Q$ is an element of a finite volume subset of $R^{3N}$, the $3N$-fold cartesian product of $R$, of volume $V^{N}$.

Notice that the continuity of the values of the positions has nothing to do with the spatial dimensionality. When we say that the continuous values of the position ${\bf r}_i$ belong to a 3D space, we mean that we need $3$ real numbers to uniquely assign a position in the space.