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Places open to the public can almost arbitrarily ask people to leave (at which point their presence becomes trespassing), they just can't ask people to leave for being members of protected classes.

So if I don't like what your t-shirt says, that's all it takes.

And for instance, Walmart doesn't have to open to the public tomorrow just because they opened to the public today.

> private public space

Please explain further what that means.

Also please explain how setting restrictions on use of the space based on conduct, such as being vaccinated or not, is more arbitrary than, say, a dress code.

> If I run a private public space I shouldn’t (and can’t) discriminate arbitrarily on who can enter.

You actually can. It might help the discussion for you to tell us where you picked up this idea.

Discrimination is largely legal. You absolutely can discriminate on any characteristic you like, with several well defined exceptions (such as race or sex).