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Zbl 0593.54036
Vermeer, J.
The smallest basically disconnected preimage of a space.
(English)
[J] Topology Appl. 17, 217-232 (1984). ISSN 0166-8641

The author shows that for every completely regular Hausdorff space X there exists the smallest basically disconnected space $\Lambda$ X which has a canonical perfect irreducible mapping onto X; i.e. there exists a perfect irreducible mapping $\Lambda$ : $\Lambda$ $X\to\sp{onto}X$ such that for every perfect irreducible mapping $g: Y\to\sp{onto}X$, where Y is basically disconnected, there exists a continuous mapping $h: Y\to\sp{onto}\Lambda X$ such that $g=\Lambda \circ h$. In the first stage of the construction the author proves that the space $\Lambda\sb 1X$ consisting of all prime prime-z-filters which are generated by open ultrafilters is homeomorphic to X iff X is basically disconnected. Next the space $\Lambda$ X is constructed as an inverse limit of a continuous inverse sequence $\{\Lambda\sb{\alpha}X,\Lambda\sp{\alpha}\sb{\beta}$; $\beta <\alpha <\omega\sb 1\}$, where $\Lambda\sb{\alpha +1}X=\Lambda\sb 1(\Lambda\sb{\alpha}X)$ for every $\alpha <\omega\sb 1$. From the existence of the space $\Lambda$ X it follows that for every locally compact basically disconnected space X there exists the smallest basically disconnected compactification BX.
[A.Błaszczyk]
MSC 2000:
*54G05 Extremally disconnected spaces, etc.
54D35 Compactifications
54C10 Special maps on topological spaces

Keywords: absolute; smallest basically disconnected space; perfect irreducible mapping; prime-z-filters

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