Baradla Barlang in Hungary actually passes through the national border into Slovakia. I believe the actual location of the border is clearly marked underground, and crossing it, you are probably entering another country unlawfully. There may even be a gate at the underground border, but I've not been there and so I am not sure. All I know is that the underground border is obvious, so there's no excuse for crossing it. All this may change now that both countries are in the EU. As far as crossing into a second property is concerned, in law the boundary of surface property extends downwards to the centre of the earth, but no further, or else we would all own part of the surface of the earth on the other side of the planet. And someone else over there might have a claim on your property. In our case, though, its only a few fish that you might have to contend with. This means that to be 100 percent sure of what you can explore, you should seek permission from more that one person. In practice, however, this can only be done if you can tell exactly where the boundary lies underground. If you can't tell that, then it's probably best to make sure that you have permission if you are using both entrances. Of course, if one owner has denied permission, then you would only be using one entrance anyway. Until someone invents an electronic tag that can be monitored through thousands of tons of rock, I can't see how anyone could tell where you were. It all boils down to acting responsibly at the end of the day.