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Perfect Mappings and Spaces of Countable Type
Published online by Cambridge University Press: 20 November 2018
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In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.
Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists
such that K ⊂ U ⊂ W. Clearly, we may assume that every member of
contains K.
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