Does downsampling increase the resolution of frequencies?
Downsampling by a factor of $N$ in time-domain means that you throw away $N-1$ samples from $x[n]$ for every $N$ samples. In frequency domain this creates $N$ shifted copies of the original spectrum and expansion of frequency axis. The shifted copies are shifted by $\omega = 2\pi \frac{k}{N}, \ k = 0,1,2,...,N-1$. So, the DTFT of downsampled sequence $x_D[n]$ is basically given by:
$$X_D(e^{j\omega}) = \frac{1}{N}\sum^{N-1}_{k=0}X\left(e^{j\left(\frac{\omega}{N} - 2\pi \frac{k}{N}\right)}\right)$$
This is why downsampling can cause aliasing and to avoid aliasing we need to Low pass filter the original sequence to suppress sprectrum outside of $|\omega| &gt; \frac{\pi}{N}$.
Now, if the original signal was appropriately oversampled, then the DTFT would have been occupying only $\omega \in [-\frac{\pi}{N}, \frac{\pi}{N}]$. And, the downsampling will expand the spectrum by a factor of $N$ without aliasing and it will give an impression that the frequency resolution has increased. Because you are now seeing the same spectrum which extended from $\omega \in [-\frac{\pi}{N}, \frac{\pi}{N}]$ stretched to $\omega \in [-\pi, \pi]$.