Mathematical Notes, Vol. 6~, No. 1, 1997 Hyperspaces of Nowhere Topologically Complete Spaces T. Banakh and R. Cauty UDC 515.12 ABSTRACT. It is proved that if X is a connected locallycontinuumwise connected coanalytic nowhere topolog- ically complete space, then the hyperspace 2 X of all nonernpty compact subsets of X is strongly universal in the class of all coanalytic spaces. Moreover, 2 X is homeomorphic to H~. if X is a BaJre space, and to Q \ HI if X contains a dense absolute Gs-set G C X such that the intersection G n U is connected for any open connected U C X. (Here HI, H2 C Q are the standard subsets of the Hilbert cube Q absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. KEY WORDS: hyperspace, connected locally continuumwise connected space, nowhere topologically complete space, coanalytic space, Z-set, absolute G6, absolute retract, meager space. For a topoldgical space X, 2x is the hyperspace of all compact subsets of X with the VietoHs topology. It was noted long ago that the geometric properties of hyperspaces are generally much better than those of the corresponding spaces. For example, D. Curtis proved that a hyperspace 2 X is an absolute retract if and only if X is a connected locally continuumwise connected space (this generalizes the results obtained by Wojdyslawski [1] and Tashmetov [2]; a space X is called locally con~inuumwise connected if for each point x E X and any its neighborhood U C X, there exists a neighborhood V of x such that for any y E V, there is a continuum lying in U and containing x and y.) Moreover, hyperspaces 2 X are sometimes homeomorphic to some well-studied model spaces of infinite- dimensional topology. An example is the classic Curtis-Schori theorem, which asserts that 2 x is homeo- morphic to the Hilbert cube Q whenever X is a nondegenerate Peano continuum. Somewhat later, Curtis proved that 2 x is homeomorphic to the Hilbert cube minus a point if X is a connected locally compact noncompact space [4], and that 2 z is homeomorphic to the Hilbert space 12 if X is a connected locally connected nowhere locally compact Polish space [5]. These three results clarify the topological structure of hyperspaces 2 X for topologically complete X and topologically classify the hyperspaces that are Polish topologically homogeneous absolute retracts: any such hyperspace is homeomorphic to one of the spaces {pt}, Q, Q \ {pt}, and s This paper studies the topology of hyperspaces 2 x for nontopologically complete spaces X. It is known that such hyperspaces 2 X are non-Borelian. Nevertheless, we prove that, sometimes, hyperspaces 2 X are homeomorphic to some well-known (although non-Borelian) model infinite-dimensional spaces such as II2n or Q \ II2n-1, even when X are not complete. The second author has recently constructed [6] subsets II,~ ( n E N) of the Hilbert cube Q, which play the same role in the classes 7)n of projective spaces as the Hilbert cube Q and the Hilbert space 12 do in the classes of metric compact and Polish spaces, i.e., they are topologically homogeneous and contain a closed copy of each space from the corresponding class. In particular, we prove that a hyperspace 2 x is homeomorphic to the space H2 if and only if X is a connected locally continuumwise connected coanalytic Baire space (Theorem 2), and that 2 x is homeomorphic to Q \ H1 if X is a connected locally connected nowhere topologically complete coanalytic space containing a dense absolute G6-set G C X such that the intersection G N U is connected for any connected open set U C X (Theorem 3). We also obtain similar results for hyperspaces homeomorphic to the spaces H2n and Q \ II2n-1 with n > 1 (see Theorem 5). 30 Translated from Ma~ema~ichesldr Zametki, Vol. 62, No. 1, pp. 35-51, July, 1997. Original article submitted April 14, 1995; revision submitted December 5, 1995. 0001-4346/97/6212-0030 $18.00 C)1998 Plenum Publishing Corporation