“…The following problem is still open: Problem 1.3. [12] Does there exist a first countable completely regular space without property (C * = C)? H. Ohta in [11] proved that the Niemytzki plane has the property (C * = C) and asked does the Sorgenfrey plane S 2 (i.e., the square of the Sorgenfrey line S) have the property (C * = C)?…”
“…The following problem is still open: Problem 1.3. [12] Does there exist a first countable completely regular space without property (C * = C)? H. Ohta in [11] proved that the Niemytzki plane has the property (C * = C) and asked does the Sorgenfrey plane S 2 (i.e., the square of the Sorgenfrey line S) have the property (C * = C)?…”
We prove that every C * -embedded subset of S 2 is a hereditarily Baire subspace of R 2 . We also show that for a subspace E ⊆ {(x, −x) : x ∈ R} of the Sorgenfrey plane S 2 the following conditions are equivalent: 2010 MSC: 54C45; 54C20.
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