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 $V=L$ there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set.Comment: preliminary versio

Haar null sets without GδG_\delta hulls

Let $G$ be an abelian Polish group, e.g. a separable Banach space. A subset $X \subset G$ is called Haar null (in the sense of Christensen) if there exists a Borel set $B \supset X$ and a Borel probability measure $\mu$ on $G$ such that $\mu(B+g)=0$ for every $g \in G$. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if $G$ is not locally compact then there exists a Borel Haar null set that is not contained in any $G_\delta$ Haar null set. We also show that $G_\delta$ can be replaced by any other class of the Borel hierarchy, which implies that the additivity of the $\sigma$-ideal of Haar null sets is $\omega_1$. The definition of a generalised Haar null set is obtained by replacing the Borelness of $B$ in the above definition by universal measurability. We give an example of a generalised Haar null set that is not Haar null, more precisely we construct a coanalytic generalised Haar null set without a Borel Haar null hull. This solves Problem GP from Fremlin's problem list. Actually, all our results readily generalise to all Polish groups that admit a two-sided invariant metric.Comment: 10 page

Characterization of order types of pointwise linearly ordered families of Baire class 1 functions

In the 1970s M. Laczkovich posed the following problem: Let $\mathcal{B}_1(X)$ denote the set of Baire class $1$ functions defined on an uncountable Polish space $X$ equipped with the pointwise ordering. $$\text{Characterize the order types of the linearly ordered subsets of $\mathcal{B}_1(X)$.}$$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 $1$ functions iff it is isomorphic to a subset of the following linear order that we call $([0,1]^{<\omega_1}_{\searrow 0},<_{altlex})$, where $[0,1]^{<\omega_1}_{\searrow 0}$ is the set of strictly decreasing transfinite sequences of reals in $[0, 1]$ with last element $0$, and $<_{altlex}$, the so called \emph{alternating lexicographical ordering}, is defined as follows: if $(x_\alpha)_{\alpha\leq \xi}, (x'_\alpha)_{\alpha\leq \xi'} \in [0,1]^{<\omega_1}_{\searrow 0}$, and $\delta$ is the minimal ordinal where the two sequences differ then we say that $$(x_\alpha)_{\alpha\leq \xi} <_{altlex} (x'_\alpha)_{\alpha\leq \xi'} \iff (\delta \text{ is even and } x_{\delta}<x'_{\delta}) \text{ or } (\delta \text{ is odd and } x_{\delta}>x'_{\delta}).$$ Using this characterization we easily reprove all the known results and answer all the known open questions of the topic

Subthreshold features of visual objects: Unseen but not unbound

AbstractThe object is a basic unit that is thought to organize the way in which we perceive and think about the world. According to theories of object-based attention, perception of unified objects depends on the binding together of the disparate features of each object via attention. Here we show that a visual feature that is not consciously perceived is nonetheless modulated by object-based attention: the influence of a subthreshold motion signal (prime) on subsequent motion perception depended critically on whether it was associated with the attended object or another, spatially overlapping object. These results show that invisibly weak features of attended objects are not lost, but are organized by and selected together with the object by attention