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. In this paper, let all maps be continuous and I = [0, 1] be the closed interval. In [BS], it was proved that the space R(I) of retractions f : I → I (i.e. f ◦ f = f) is homeomorphic (≈) to the pseudo-interior s = (−1, 1) of the Hilbert cube Q = [−1, 1], where R(I) has the supmetric. Thus the Hilbert cube Q is a compactification of R(I). Here we consider such a natural compactification of R(I). Equip the product I = I× I with the following metric: d((x, y), (x′, y′)) = max{|x− x′|, |y − y′|}, and let exp(I) be the hyperspace of nonempty compact subsets of I endowed with the Hausdorff metric:

A Hilbert cube compactification of the space of retractions of the interval