Is there a chemical-free way to keep pool water safe and healthy, that is without chlorine or bromine?
How about using UV radiation to sterilize the water?
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Still chlorinated, but if handling chemicals is the issue, how about a salt water pool?
The alternative I've seen pushed is copper-oxygen system. I can't say how effective it is, but last I checked it is not approved by the EPA for sanitizing water. I've also read that they take more attention and care.
Just based on experience having a pool in full sunlight - UV isn't going to do the job by itself. I think the filters they sell are designed to reduce your chemical dependency, but not eliminate it. Same with ozone generators.
Ozone treatment is another option. I have swum in two ozone pools - the two obvious benefits:
This site has some info: http://www.lenntech.com/library/ozone/swimming-pool/ozone-applications-swimming-pool-water.htm