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Extremely metal-poor (EMP) stars are the living fossils with records ofchemical enrichment history at the early epoch of galaxy formation. By therecent large observation campaigns, statistical samples of EMP stars have beenobtained. This motivates us to reconsider their classification and formationconditions. From the observed lower-limits of carbon and iron abundances ofAcr(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 isindispensable for the fragmentation of their parent clouds to form suchlow-mass, i.e., long-lived stars, and that the dominant grain species arecarbon and silicate, respectively. We constrain the grain radius r_i^{cool} ofa species i and condensation efficiency f_{ij} of a key element j asr_C^{cool}/f_{C, C} = 10 um and r_Sil^{cool}/f_{Sil, Mg} = 0.1 um to reproduceAcr(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 theconventional boundary [C/Fe] = 0.7 between CE- and CN-EMP stars, this conditionsuggests a physically meaningful boundary [C/Fe]_{b} = 2.30 above and belowwhich carbon and silicate grains are dominant coolants, respectively.

Classification of extremely metal-poor stars: absent region in A(C)-[Fe/H] plane and the role of dust cooling