Consider the fact that a real algebraic set that is irreducible in Zariski topology can be even disconnected. We define a notion of irreducibility based on arc-analytic functions, which leads to a finer decomposition of, more generally, semialgebraic sets. An application is that irreducibility in this sense characterizes the semialgebraic sets on which the Identity Principle is true for every arc-analytic function. In fact, pairing of such functions with semialgebraic sets leads to a standard theory; for instance, Nullstellensatz holds too. An essential prerequisite to development of this theory was the fact that every arc-symmetric set is the zero locus of an arc-analytic function, which had been conjectured by Kurdyka in 1988. We shall sketch a proof of this conjecture as well.