This paper is devoted to a natural generalization of the problem on the chromatic number of the plane. The chromatic number of the spaces ℝⁿ × [0, ε]^k is considered. It is proved that 5 ≤ χ(ℝ² × [0, ε]) ≤ 7 and 6 ≤ χ(ℝ² × [0, ε]²) ≤ 7 for ε > 0 sufficiently small. Also, some natural questions arising from these considerations are posed.