Basically, yes. Copper will corrode (oxidize) becoming copper ions in water that can eventually stain. How fast this occurs depends on a variety of factors. The pH is most important with low pH rapidly increasing corrosion. Higher chlorine levels is another factor with the FC/CYA ratio being most relevant (or FC alone, if CYA is near zero). The conductivity of the water, roughly approximated by TDS or salt level, is another factor.

So a particularly corrosive scenario would be a pool without CYA (such as most indoor pools) with an SWG system (so the water has 3000 ppm salt) operating at a lower pH. Nevertheless, it would be best to avoid the use of pure copper in pools.

Copper pipes in homes have been exposed to chlorine (up to 1 ppm FC) with no CYA and have generally not corroded, but there is little dissolved oxygen in the pipes, there is little conductivity since the TDS is low, and there are some corrosion inhibitors usually added to the water.

Richard