Purely infinite C*-algebras of real rank zero

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We show that a separable purely infinite C*-algebra is of real rank zero if and only if its primitive ideal space has a basis consisting of compact-open sets and the natural map K₀(I) -> K₀(I/J) is surjective for all closed two-sided ideals J contained in I in the C*-algebra. It follows in particular that if A is any separable C*-algebra, then A tensor O₂ is of real rank zero if and only if the primitive ideal space of A has a basis of compact-open sets, which again happens if and only if A tensor O₂ has the ideal property, also known as property (IP).