On the converse of Tietze-Urysohn's extension theorem.

Abstract

From the text: "Problem A: Characterize (normal) spaces in which every C -embedded subset is closed. Problem B: Characterize (normal) spaces in which every C ∗ -embedded subset is closed. Our aim here is to call attention to the above problems, and provide some partial results in this line. Question C: Suppose that X and Y are completely regular spaces in which every C -embedded subset is closed. If C(X) is isomorphic to C(Y) , is then X homeomorphic to Y ? Question D: Suppose that X and Y are completely regular spaces in which every C ∗ -embedded subset is closed. If C ∗ (X) is isomorphic to C ∗ (Y) , is then X homeomorphic to Y ?''