  Extremely metal-poor (EMP) stars are the living fossils with records of
chemical enrichment history at the early epoch of galaxy formation. By the
recent large observation campaigns, statistical samples of EMP stars have been
obtained. This motivates us to reconsider their classification and formation
conditions. From the observed lower-limits of carbon and iron abundances of
Acr(C) ~ 6 and [Fe/H]cr ~ -5 for C-enhanced EMP (CE-EMP) and C-normal EMP
(CN-EMP) stars, we confirm that gas cooling by dust thermal emission is
indispensable for the fragmentation of their parent clouds to form such
low-mass, i.e., long-lived stars, and that the dominant grain species are
carbon and silicate, respectively. We constrain the grain radius r_i^{cool} of
a species i and condensation efficiency f_{ij} of a key element j as
r_C^{cool}/f_{C, C} = 10 um and r_Sil^{cool}/f_{Sil, Mg} = 0.1 um to reproduce
Acr(C) and [Fe/H]cr, which give a universal condition 10^{[C/H]-2.30} +
10^[Fe/H] > 10^{-5.07} for the formation of every EMP star. Instead of the
conventional boundary [C/Fe] = 0.7 between CE- and CN-EMP stars, this condition
suggests a physically meaningful boundary [C/Fe]_{b} = 2.30 above and below
which carbon and silicate grains are dominant coolants, respectively.
