Hmm... Not sure about that one.

But the ability to fill would imply the ability to empty though.

Most people see empty as a mixture of nitrogen, oxygen, etc. (Aka: air) Not 'void of any atoms'..

No because the empty set is part of null or nothing

I'd argue that depends on what the definition of empty is. If you say "something that contains nothing", I would argue the fact you say "contains" implies it has the ability to hold something.

However the ability to be full may not be there. If we're being theoretical you could have something of infinite volume.

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there is a lot of "empty" space within an atom, but ignoring edge cases like a neutron star, etc. it's not like other material can occupy that space, so is it really empty

Well technically it's empty but as far as i know from watching some interviews in the past  is that's supposed to be pure electron field forces  repelling between those atoms that create that empty space. kinda like opposing poles in magnetism . an example which said was like you sitting on a chair technically you're making contact with the chair as you sit but if you get in on it close like zooming in real close to a level of atoms between you & the chairs you're basically sitting as in hovering on empty space being repelled purely by forces repelling between atoms in the chair & atoms that are in you . that empty space is exactly like space between two magnets repelling each other.

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Does the ability to be empty imply and require the ability to be full?

with that said i have no idea how this would work out.