Topless tanning, like any tanning in general, in no way affects the risk of breast cancer. Ultraviolet radiation does not reach the breast tissue, while heating of the surface tissues (skin and superficial subcutaneous tissue) during the tanning process does not affect the risk. But of course we have to think of uv protection to reduce the risk of melanoma.

Exactly the same applies to tanning in a solarium with a stikini. This has nothing to do with breast cancer, but it can really save you from an ultraviolet burn of the nipple's skin.