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ON SPACES WITH DIRECTED SYSTEMS OF OPEN MAPS

Annotation

If a topological space X has a countably directed system L of open maps fα onto hereditarily separable spaces Xα, α  A , and L has some natural properties, then 1. any system λ of L –cylinder subsets in X (i. e. sets of the form f−1αM such that M ⊂ Xα, α ∈ A ) and any system λ of unions of G -subsets in X contains a countable subsystem μ of λ , that is dense in λ (i.e. the closures of the unions of the systems μ and λ coincide), 2. any map f of X onto a separable metric space may be «passed through» one of maps fα (i.e. f is the composition of this fα and a map of Xα ). These results allow to obtain correspondent results for Tychonoff products of hereditarily separable spaces and for subspaces of Tychonoff products of spaces with a countable net.

Keywords

countably directed system of open onto maps of a topological space; Tychonoff product; S -product; hereditarily separable space; space with a countable net; G -(Suslin property)

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UDC

515.12

Pages

1068-1071

Received

2015-05-22

Section of issue

Scientific articles

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