On homogeneous totally disconnected 1-dimensional spaces

Abstract

On homogeneous totally disconnected 1-dimensional spaces by Kazuhiro K a w a m u r a (Saskatoon), Lex G. O v e r s t e e g e n (Birmingham, Ala.) and E. D. T y m c h a t y n (Saskatoon)

On cohesive almost zero-dimensional spaces

“…For the spaces E, E c , and E ω c there is a stronger result: in these spaces σ-compacta are negligible, see [13], [23], and [11].…”

“…In [32, §6 Example p.596], Nishiura and Tymchatyn implicitly proved that D e , the set of endpoints of Lelek's fan [27, §9], is not rational at any of its points. Results in [5,6,23] later established D e E c , so E c is nowhere rational. Working in 2 , Banakh [3] recently demonstrated that each bounded open subset of E has an uncountable boundary.…”

A characterization of the product of the rational numbers and complete Erdős space

“…As mentioned in the introduction, E c is a cohesive almost zero-dimensional space. An extrinsic characterization of E c is given by Lelek functions as follows: if [7]. An intrinsic characterization of E c was given in [2].…”

“…As mentioned in the previous section, by the extrinsic characterization of E c from [7], in Definition 3.1 we will have that G ϕ↾Xn 0 is homeomorphic to E n for each n ∈ ω. So indeed, E is a countable increasing union of nowhere dense subsets, each homeomorphic to complete Erdős space.…”

“…It was soon noticed that some interesting Polish spaces are homeomorphic to E c , see [7]. Due to the interest in these two spaces, Jan Dijkstra and Jan van Mill obtained topological characterizations of E c and E (see [2] and [3], respectively), and applied them to show that some other noteworthy spaces are homeomorphic to either one of these two.…”