The small boundary property in products

For a continuous action G X of a countable group on a compact metrizable space we show that the following are equivalent: (i) the action G X has the small boundary property and no finite orbits, (ii) for every continuous action H Y of a countable group on a compact metrizable space, the product action G× H X× Y has the small boundary property. In particular, (ii) is automatic when G is infinite and the action G X is minimal and has the small boundary property. The argument relies on a small boundary version of the Urysohn lemma.