Topological Properties of the Hilbert Cube and the Infinite Product of Open Intervals

1. For each i>0, let 7, denote the closed interval O^x^ 1 and let °I¡ denote the open interval 0<*< 1. Let 7°° =fli>o 7 and °7°° = rii>o °h7°° is the Hilbert cube or parallelotope sometimes denoted by Q. °7°° is homeomorphic to the space sometimes called s, the countable infinite product of lines. The principal theorems of this paper are found in §§5, 7, 8, and 9. In §5 it is shown as a special case of a somewhat more general theorem that for any countable set G of compact subsets of °7°°, °7eo\G* is homeomorphic to °7°° (where G* denotes the union of the elements of G)(1). In §7, it is shown that a great many homeomorphisms of closed subsets of 7" into 7" can be extended to homeomorphisms of 7TM onto itself. The conditions are in terms of the way in which the sets are coordinatewise imbedded in 7°°. A corollary is the known fact (Keller [6], Klee [7], and Fort [5]) that if jf is a countable closed subset of 7°°, then every homeomorphism of A" into 7 e0 can be extended to a homeomorphism of 7°° onto itself. In a further paper based on the results and methods of this paper, the author will give a topological characterization in terms of imbeddings of those closed subsets X of 7 e0 for which homeomorphisms of X into Wx={p \pelx and the first coordinate of p is zero} can be extended to homeomorphisms of X onto itself. In his recent dissertation, Raymond Wong has settled a question of Blankinship [4] by showing that there do exist two Cantor Sets in 7" such that no homeomorphism of one onto the other can be extended to a homeomorphism of 7°° onto itself. In §8 the results of §7 are used to give conditions under which the union of two Hilbert cubes can be seen to be homeomorphic to 7°°. In §9 it is proved that many countable infinite products not obviously homeomorphic to °7C0 are, in fact, homeomorphic to °7°°. A theorem (Theorem 9.5) equivalent to the following is proved: "For i= 1, 2,..., let C¡ be a closed nrcell