Unequivocally, yes, AC is independent of wielded weapons.
Unless any of the following are true:
- They have a feat that grants them extra AC in certain circumstances that is dependent on a weapon (e.g. Dual Wielder or Defensive Duelist)
- They have a magic weapon that adds to their AC. (Staff of Defense)
As Theo says, AC measures so many more things than the ability to block an attack. For instance, many characters use their DEX as part of their AC, indicating that their ability to dodge attacks is more important than their ability to block or absorb them. Note that a bow wielding character's AC is the same as a sword wielder. If it was meant to factor in your weapon, you'd get an advantage with certain kinds of weapons.
This brings us to an important point. You can rule differently on this. Though I'm not sure you'd want to. Remember that D&D is not intended to be a RL combat simulator. It's a game with many abstractions. If you want to remove one in favor of a more verisimilitudinous approach, that's probably fine. However, you will need to be consistent across the whole subsystem. You're going to need to bolt on rules for how a weapon aids AC. Does it simply prevent an opponent from having advantage against you? Would there be a weapon that can do one better and inflict disadvantage? Does it provide some kind of mysterious bonus to AC (perhaps what takes you to a starting 10?), would not having a weapon and wearing cloth with a Dex of 0 only mean an 8 AC. These are the kinds of questions you should ask, and more, study hard, before you make a ruling on something like this that has pretty broad implications.