Once water is boiled to remove oxygen does it remain oxygen free when cooled?

As soon as the water begins to cool off, it begins to reabsorb oxygen again. By the time it has returned to room temperature, it will contain significant amounts of oxygen. If you want to maintain the water in a degassed state, right after boiling it you should store it in a pyrex bottle that has been filled up as close to the cap as possible, and screw the cap on tightly.