$$X(e^{j\omega}) = \sum_{n=-\infty}^\infty x[n] e^{-j\omega n} $$
The frequency term $\omega$ in DTFT is normalized as $\omega = \frac{\Omega}{f_\mathrm{s}}$
$\Omega= 2 \pi f$ is the angular frequency continuous-time Fourier Transform of $x(t)$.
A continuous-time signal that can be properly sampled will have non-zero energy components in its spectrum having frequency $|f| \lt \frac{f_\mathrm{s}}{2}$
Hence the range of $|\omega|$ should lie between $0$ through $\pi$. Given that the spectrum of a discrete time signal repeats every $2\pi$ radians, we should see this spectrum replicated from 0 to $\pi$, $2\pi$ to $3\pi$, $4\pi$ to $5\pi$ and so on.
I can extrapolate this to negative factors of $2\pi$ as well. So the spectra can additionally replicate from $-2\pi$ to $=3\pi$, $-4\pi$ to $-5\pi$ and so on.
I have however read that the range of $\omega$ is from $-\pi$ to $+\pi$. Where does this come from?