The space of infinite-dimensional compacta and other topological copies of \((l_ f^ 2)^ \omega\). (English) Zbl 0786.54012

Dijkstra, J. J.; van Mill, Jan; Mogilski, J.

Pac. J. Math. 152, No. 2, 255-273 (1992).

Summary: We show that there exists a homeomorphism from the hyperspace of the Hilbert cube \(Q\) onto the countable product of Hilbert cubes such that the \(\geq k\)-dimensional sets are mapped onto \(B^ k\times Q\times Q\times\cdots\), where \(B\) is the pseudoboundary of \(Q\). In particular, the infinite-dimensional compacta are mapped onto \(B^ \omega\), which is homeomorphic to the countably infinite product of \(l^ 2_ f\). In addition, we prove for \(k\in\{1,2,\dots,\infty\}\) that the space of uniformly \(\geq k\)-dimensional sets in \(2^ Q\) is also homeomorphic to \((l^ 2_ f)^ \omega\).

MSC:

54B20	Hyperspaces in general topology
54E45	Compact (locally compact) metric spaces

Keywords:

Hilbert cube