139 research outputs found
Transfinite inductions producing coanalytic sets
A. Miller proved the consistent existence of a coanalytic two-point set,
Hamel basis and MAD family. In these cases the classical transfinite induction
can be modified to produce a coanalytic set. We generalize his result
formulating a condition which can be easily applied in such situations. We
reprove the classical results and as a new application we show that in
there exists an uncountable coanalytic subset of the plane that intersects
every curve in a countable set.Comment: preliminary versio
Characterization of order types of pointwise linearly ordered families of Baire class 1 functions
In the 1970s M. Laczkovich posed the following problem: Let
denote the set of Baire class functions defined on an
uncountable Polish space equipped with the pointwise ordering.
The main result of the present paper is a complete
solution to this problem.
We prove that a linear order is isomorphic to a linearly ordered family of
Baire class functions iff it is isomorphic to a subset of the following
linear order that we call , where
is the set of strictly decreasing transfinite
sequences of reals in with last element , and , the so
called \emph{alternating lexicographical ordering}, is defined as follows: if
, and is the minimal ordinal where the
two sequences differ then we say that Using this characterization we easily reprove all
the known results and answer all the known open questions of the topic
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