A note on nonfragmentability of Banach spaces.
We show that AC is equivalent to the assertion that every compact completely regular topology can be extended to a compact Tychonoff topology.
The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.
For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an -extender (resp. -extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous -extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and...
An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C*-embedded subspaces of ω*, showing that there can be such sets of cardinality and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.