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Research papers and publications
## Open Problems

The following problems are open, at least for me. The first person who
provides me a correct complete solution (preferrably by
e-mail) can be awarded
by the mentioned prize. As a unit I use, according to Czech tradition, one
beer. This means either one usual glass of Czech beer or one glass of Czech
wine or one glass of apple juice. The prize may be received only in
the Czech Republic.
**Definition.** A compact space *K*
is called *Valdivia* if it is homeomorphic to some
such that
is dense in *K'*.

**PROBLEM 1** *(2 beers)*
Is there a scattered Valdivia compactum containing a copy of
?

A topological space is *scattered* if any its nonempty subset
has a relatively isolated point.

**PROBLEM 2** *(Asked at 29th Winter School in Lhota nad Rohanovem,
February 2001)* Let
and .
Is the dual unit ball of *Y*, in its weak* topology,
a Valdivia compactum?

Possible answers:

a) No. *(3 beers)*

b) Yes, and the homeomorphism in the definition of a Valdivia compactum
may be chosen linear. *(2 beers)*

c) Yes, but the homeomorphism cannot be chosen linear. *(10 beers)*

**PROBLEM 3** *(3 beers)* Let *K* and *L*
be nonempty compact spaces such that
is Valdivia.
Are *K* and *L* Valdivia, too?

**Remark:** Yes, if each one of them has at least one
-point (easy) or if at least one
of them has a dense set of -points
(not easy).

CESKY - ISO 8859