That analogy would be more accurate if you also operated a cafe out of your living room, with a big "open" sign on the front door, and someone accidentally used

That analogy would be more accurate if you also operated a cafe out of your living room, with a big "open" sign on the front door, and someone accidentally used your personal bathroom because you failed to stick a "private" sign on it. In that case, it would be unreasonable to sue someone for trespass.