By Y. K. ZAHAROY and A. 1'. K o r . ~r x o r of Leningrad (R,eceired March 27. 1981)
In [ i ] and [2] the notion of u-cover of a compact space was introduced. A compact space a> is called a u -c o w of c( coinpnct spcrce S iff it is the smallest basically disconnected ( = a-extremally disconnected) perfect irreducible preimage of S . Rut the property of "heingthesmallest" is not an inner property of the pair a&' and t,: a,S -8. This paper presents the inner characterization of a 2 using the lifting of RAIRE sets and RAIRE functions from S t o a 3 (Theorems 2 and 4). For that purpose some general construction of a G-cover of a compact space S is considered whose particular cases are a,S and aS (the absolute of S ) .
Theorems 2 and 4 present the characterization of any G-cover from which (in a particular case) the ahove characteristic of the o-cover a,$ follows. In another particular case the characteristic of the GLEASON cover (absolute) a 8 IS ol)tainetl using the lifting of BOREL sets and BOREL functions from IS to US.
Appling Theorems 2 and 4 and the fact that a$ is smallest u-extremall> dis-( onnected preimage of S we get the following characterization of the Booman alpelm go(&!) of equiralence classes of all the R A t m sets and of the vector lattice B,(S) of equivalence classefi ofhi1 the R A ~R P functions on 8 : a,(S) is the smallest o-complete Rooman algelra containning the lattice of all the cozero-sets 111 S. and B,(S) ib the smallest U-DEDEXIND complete vector lattice containing the vector lattice C(S). But since the properties of such classical ohjects a RAIRE sets and RAIRIC functions are interesting in them~elves the direct proof without using the jiroperties of a a$ of the a h v e characterization is presented (Theorems 1 and 3).