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A dichotomy theorem for the Ellentuck topology. (English) Zbl 1065.03029

Summary: Results are deduced from the following dichotomy which reduces descriptive properties concerning the Ellentuck topology to two well-known examples. Theorem: Every nonempty perfect set in the Ellentuck topology contains a closed copy of the Sorgenfrey line or a closed copy of the rational numbers. This leads to a Marczewski-Burstin representation for Marczewski sets in the Ellentuck topology.

MSC:

03E15 Descriptive set theory
03E20 Other classical set theory (including functions, relations, and set algebra)
28E15 Other connections with logic and set theory
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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