When someone is said to be the owner of a piece of land do they own it all the way down to the core of the earth and as high as the sky?

Ownership applies only to the surface area. Any further claims would require mineral and water rights. no one can own the sky or air. Philosophically, we are all just stewards of the land anyhow. The size of the land that is owned is determined by the land survey/plot dimensions documented in conjunction with the title/warranty deed to the property. The landowner also owns mineral, water rights and all other such accoutrements as determined in the original purchase contract and laws of the state where the land is located. The exception would be any issues unless that are specifically contraindicated by the contractual agreement itself or another legal instrument validated by the court. In fact air space is a matter of property. This is particularly true in developed cities such as New York where air rights over various structures and other developments are regularly bought and sold. Probably the most famous air right was over the Pennsylvania Railroad tracks on the West side of Manhattan in (not sure of the correct preposition) which Lincoln Center was built. The issue of mineral rights is not straight forward either. For example if an owner has dug a mine on her property and found a seam of precious metal ore she has the right to continue to mine that seam even if it passes under someone surface right or into the mineral right of another owner. In addition there are separate rights associated with surface water-- above and beyond using the water for agricultural or domestic purposes--that runs across a surface that are referred to as riparian rights. I am not a lawyer or other expert but since lawyers have been dealing with these matters for close to five thousand years we have a better chance of understanding the theory of relativity than property law. "Ownership applies only to the surface area. " That statement is inaccurate. In my section of the country you own the land and what's under it unless the prior owner reserved some mining rights when she conveyed the land to you. If you buy 100 acres and it turns out that there's a huge gravel deposit underneath- it's all yours.