Yes, there are some special membranes that make this possible. These membranes let o-h radicals pass through which are literally water molecules. But will not let O2 pass through. I have tested myself these air tight membranes in the lab.
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Yes. I have a bottle of water with a different cap and it doesn't leak water yet when I flip it over and squeeze only air comes out never water. I've squished the bottle and when the air is all out the bottle won't squish anymore.
No. Air tight means that no air can pass through and water may be able to, but water tight means that no water can pass through and air may be able to.