### A note on nonfragmentability of Banach spaces.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

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The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his.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 ${L}_{ch}$-extender (resp. ${L}_{cch}$-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 ${L}_{ch}$-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 $c$ and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.