Abstract: Some key tools, such as the Nullstellensatz, of course fail in algebraic geometry over the reals. However, having less of such machinery that works universally (as in the case over an algebraically closed field) does not make real algebraic geometry any less fascinating. One needs instead to come up with the right tools that are tailored to a phenomenon of interest. We are interested in 'irreducible components' of real algebraic sets, particularly as they relate to the sheets of Nash on such sets. As it turns out, we need to extend the classical ring of polynomials and obtain a wider class of functions to work with. We compare different approaches and present our point of focus, namely the ring of arc-analytic functions, in which even the Nullstellensatz is proven to hold.