You can keep your hat on?

OK, so now I'm left suffering from an existential uncertainty over whether hats, and beyond that, headwear in general, are not considered clothing.

I mean, this sign seem to be pretty damn firm about the policy that no clothes are allowed. None at all. Without exception. That's extremely unambiguous and doesn't leave any wiggle room.

But, at the same time, it shows a nude woman wearing a hat! So, is that to say that hats are not clothes? Because that would be a pretty major revelation for me, if it's true! It would leave me wondering whether I need to reclassify other wearables and that is a rabbit hole that I have no desire to get lost down.

What do you think? Are hats clothes, or are they something else entirely? Let me know!