Method of inverse spectra in topology of bicompacts

AbstractWe introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk, 1976, 31(5), 191–226 (in Russian)] consisting of skeletal maps.

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TOPOLOGY OF LIMIT SPACES OF UNCOUNTABLE INVERSE SPECTRA

Russian Mathematical Surveys ◽

10.1070/rm1976v031n05abeh004195 ◽

1976 ◽

Vol 31 (5) ◽

pp. 155-191 ◽

Cited By ~ 24

Author(s):

E V Shchepin

Keyword(s):

Inverse Spectra

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Finite approximation of stably compact spaces

Applied General Topology ◽

10.4995/agt.2002.2063 ◽

2002 ◽

Vol 3 (2) ◽

pp. 197 ◽

Cited By ~ 4

Author(s):

M.B. Smyth ◽

J. Webster

Keyword(s):

Function Space ◽

Directed Graphs ◽

New Method ◽

Hausdorff Spaces ◽

Topological Graphs ◽

Topological Approach ◽

Compact Spaces ◽

Finite Directed Graphs ◽

Finite Approximation ◽

Inverse Spectra

<p>Finite approximation of spaces by inverse sequences of graphs (in the category of so-called topological graphs) was introduced by Smyth, and developed further. The idea was subsequently taken up by Kopperman and Wilson, who developed their own purely topological approach using inverse spectra of finite T<sub>0</sub>-spaces in the category of stably compact spaces. Both approaches are, however, restricted to the approximation of (compact) Hausdorff spaces and therefore cannot accommodate, for example, the upper space and (multi-) function space constructions. We present a new method of finite approximation of stably compact spaces using finite stably compact graphs, which when the topology is discrete are simply finite directed graphs. As an extended example, illustrating the problems involved, we study (ordered spaces and) arcs.</p>

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