How does $s$ subshell not have a node in the center despite the nucleus being there? In most images of $1s$ subshell I see that there's no node shown at the center, and even the formula $n-\ell-1$ gives 0 as the answer.
But, isn't the nucleus experimentally proven to be at the center? And as the mass should be much more than individual electrons, I'm assuming the nucleus doesn't have large probability clouds like individual electrons do - i.e. the nucleus is mostly at the center.
If my assumption is correct, then shouldn't it be impossible for the electron cloud to occupy any space at the center as it's already taken up? Even for the other orbitals there seems to be empty space in the center, so why not for the $s$ orbital?
I did see one image of the $s$ orbitals where it appeared like they showed a node but that was just one image, every text or any other image says otherwise. So, could someone verify and explain whether there is a node or not?
 A: These subshells are just the Spherical Harmonics. As you can see from that Wikipedia article, the S orbitals do not have a node in the center.
There is no reason why an electron couldn't be "inside" a nucleus. As long as particles differ in some "quantum number", there is no physical law that would forbid that. For example, an electron and a proton are different. Or two electrons with different spin.
It is that for that very reason, namely that S-orbital electrons do have a finite probability of being within the nucleus, that some isotopes decay via electron capture.
A: The idea that 'space is already taken up' is a classical concept which one has to learn to drop. The nucleus and the electron can be at the same point - or, equivalently, if you consider a small volume $dV$ at the origin, the probability that both the nucleus and the $s$ electron are in that volume tends to some finite number times $dV$ as $dV$ tends to zero.
Classical objects can't exist in the same space because of the overlapping electron wavefunctions and the Pauli principle. At the atomic level objects can occupy the same space - and they do.
