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A note on zero-dimensional preimages of compact spaces. (English) Zbl 0643.54025

The author uses an inverse limit construction to prove that “every compact Hausdorff space of power continuum is a continuous image of a zero dimensional compact space of power not greater than \(\Sigma \{2^{\tau}:\) \(t<2^{\omega}\}\). Moreover, if the character at every point of the given space is less than the continuum, then the resulting space has the same property”. It follows that under CH every first countable compact space is the continuous image of a zero dimensional first countable compact space. This gives a partial answer to a question of E. van Douwen. As the author notes in a comment added in proof, the question was also asked by V. I. Ponomarev and partially answered by A. V. Ivanov [Usp. Mat. Nauk 35, No.6(216), 161-162 (1980; Zbl 0458.54021); Engl. translation in Russ. Math. Surveys 35, No.6, 95-96 (1980)]. Ivanov also uses an inverse limit construction, but quite differently. Another proof of this result has recently been constructed by V. V. Tkachuk [Commentat. Math. Univ. Carol. 29, No.1, 73-78 (1988)]. Tkachuk has also shown that CH may be eliminated if the original space is Eberlein compact or a first countable compact LOTS.
Reviewer: L.M.Friedler

MSC:

54D30 Compactness

Citations:

Zbl 0458.54021
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