COLLOQUIUM MATHEMATICUM VOL. * 200* NO. * ON UNIVERSALITY OF FINITE POWERS OF LOCALLY PATH-CONNECTED MEAGER SPACES BY TARAS BANAKH (Lviv and Kielce) and ROBERT CAUTY (Paris) Abstract. It is shown that for every integer n the (2n + 1)th power of any locally path-connected metrizable space of the first Baire category is A 1 [n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X n+1 is A 1 [n]-universal for every n. A topological space X is called C -universal , where C is a class of spaces, if X contains a closed topological copy of each space C ∈C . We denote by M 0 , M 1 , and A 1 the classes of metrizable compacta, Polish spaces, and metrizable σ-compact spaces, respectively. For a class C of spaces we denote by C [n] the subclass of C consisting of all spaces C ∈C with dim C ≤ n. In terms of universality, the classical Menger–N¨ obeling–Pontryagin– Lefschetz Theorem states that the cube [0, 1] 2n+1 is M 0 [n]-universal for every n ≥ 0. It is well known that the exponent 2n + 1 in this theorem is the best possible: the Menger universal compactum μ n cannot be embedded into [0, 1] 2n . Nonetheless, P. Bowers [Bo] has proved that the (n +1)th power D n+1 of any dendrite D with dense set of end-points is M 0 [n]-universal for every non-negative integer n. Moreover, every such dendrite D contains a connected G δ -subset G whose (n + 1)th power G n+1 is M 1 [n]-universal for every n (see [Bo]). Actually, these results of Bowers’ are particular cases of a more general fact proved in [BCTZ]: for any locally connected Polish space X without free arcs the power X n+1 is M 0 [n]-universal; moreover, if the space X is nowhere locally compact, then the power X n+1 is M 1 [n]-universal. Taking into account that M 0 and M 1 are the first classes in the Borel hierarchy it is natural to ask the following Question. Suppose C is a Borel class. Is there a one-dimensional abso- lute retract X ∈C whose (n + 1)th power X n+1 is C [n]-universal for every integer n ≥ 0? 2000 Mathematics Subject Classification : 4F45, 54B10, 54F50, 54H05, 54C55, 54D45, 57N20. Key words and phrases : locally path-connected, σ-compact, universal, meager. [1]