Gosh...
Public airfields are just that, and are freely available for all to use.
Yes, if they are "open". EVERY public airfield has an authority which operates it - person, committee, council, whatever. That entity can declare the airfield closed any time they want. Or, I as a firefighter I can, if I'm attending to an accident there. If the airfield is declared closed by an authority [which could include emergency services], it is closed, and landing there would be at least trespassing, if not an offense.
I have personally closed a public airfield, by phoning flight service, identifying myself as a firefighter, and instructing a notam be issued closing the airfield due to a fatal accident. Then I parked a fire truck in the middle of the runway, for the benefit of those who don't check notams. That airfield was then closed - no negotiation. The airfield "authority" did not yet know that I had closed the airfield, but it was closed. Indeed, he never found out that I closed the airfield, as
he was the fatality. If a pilot had landed there during the closed time, I would have had that pilot violated.
Some on here seem to infer a private airfield is closed when in fact all that is needed is the owners permission to use that strip. This was the point of confusion.
A private airfield is closed if the owner does anything which indicates it is closed. In absence of closing action on the owner's part, it is only "open" if you have the owner's explicit permission to operate there then. Otherwise, you are trespassing, though happily, most private airfield operators are gracious about this.
It would be presumptuous to assume that you can land wherever you like (trespass) without assuring the "open" [to you] status of the airfield. Sometimes people get away with presumptuous, but it's still presumptuous, it's not "open".
My 2 cents worth, as a private airfield owner, and formally, an employee of a public airfield operator.