It’s not a public space, it’s a publicly-accessible space. Limits on public access can be imposed by the property owners using trespass law. Violating posted signs is trespassing; cops that enter without a warrant or other special authority (beyond just being a cop, eg hot pursuit) are subject to criminal trespass.
It’s a public space. The limitations on search warrants are very low for those.
It’s not a public space, it’s a publicly-accessible space. Limits on public access can be imposed by the property owners using trespass law. Violating posted signs is trespassing; cops that enter without a warrant or other special authority (beyond just being a cop, eg hot pursuit) are subject to criminal trespass.