Devise a method to obtain pure water from seawater

Distillation works. Reverse osmosis is another proven technology.

There is a type of igneous rock that acts as a filter. They often use it to filter gasses. Several different types of it occur. It can also be man made. The holes in the rock can be such to allow a smaller molecule through but not a larger one. Maybe this could be applied for the same application.

The pressure needed is fairly large but a system could be devised to create a free pressure system... say along the panama canal. Gravity would do all the dirty work. If I find the name of it I will post it here for you.

Oxygen Concentrators

I was reading about this and a few other things. The rock they talk about here is Zeolite. It is one in a class or many others that perform similarly.

A method for desalinizing seawater could be easily adapted to this method

There are many zeolites. It's not really a "rock". These substances are often called "molecular sieves" and are characterized the molecular pore size. They will absorb sodium or chlorine, if the chambers are the correct size. This wouldn't be very economical.