Document Zbl 0885.54020

Van Douwen’s problem on 0-dimensional images of ordered compacta. (English) Zbl 0885.54020

Proc. Am. Math. Soc. 126, No. 1, 269-271 (1998).

Summary: A compact 0-dimensional space has a \(T_0\)-separating rank 1 family of clopen sets iff it is the continuous image of a compact 0-dimensional linearly ordered space.

Cited in 1 Document

MSC:

54D70	Base properties of topological spaces
54F05	Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54C05	Continuous maps
54D30	Compactness
54F50	Topological spaces of dimension \(\leq 1\); curves, dendrites
06A05	Total orders

Keywords:

0-dimensional space; continuous image; linearly ordered compactum; \(T_ 0\)-separating family; rank 1 family; Boolean algebras; dendron

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References:

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[4]	Sabine Koppelberg and J. Donald Monk, Pseudo-trees and Boolean algebras, Order 8 (1991/92), no. 4, 359 – 374. · Zbl 0778.06011 · doi:10.1007/BF00571186
[5]	Jan van Mill and Evert Wattel, Dendrons, Topology and order structures, Part 1 (Lubbock, Tex., 1980) Math. Centre Tracts, vol. 142, Math. Centrum, Amsterdam, 1981, pp. 59 – 81. · Zbl 0469.54017
[6]	Jacek Nikiel, Orderability properties of a zero-dimensional space which is a continuous image of an ordered compactum, Topology Appl. 31 (1989), no. 3, 269 – 276. · Zbl 0683.54037 · doi:10.1016/0166-8641(89)90023-0
[7]	P. J. Nyikos, personal communication.
[8]	S. Purisch, Problems Section, Top. Proc. 17 (1992), 412-413.

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