On compacta which are $l$-equivalent to $I^{n}$

Abstract

All spaces considered in this paper are assumed to be metrizable. A compactum is a compact space. A continuum is a connected compactum, and a mapping is a continuous function. For a space X we denote by C(X) the space of all real-valued mappings on X with the topology of uniform convergence. Then by Milutin's interesting work [8], we have known that for each pair of uncountable compacta X and Y, C{X) is linearly isomorphic to C(Y) (see [12] for the detailsand generalizations). On the other hand, for space X we denote by CP(X) the space of all real-valued mappings on X with the topology of pointwise convergence. Spaces X and Y are said to be l-equivalent[1] provided that CP(X) is linearly isomorphic to CP(Y), written CP(X) = CP(Y). Recently, Pavlovskii [11] showed the following.