Projective Fraïssé limits and the pseudo-arc

Abstract

Abstract. The aim of the present work is to develop a dualization of the Fraïssé limit construction from model theory and to indicate its surprising connections with the pseudo-arc. As corollaries of general results on the dual Fraïssé limits, we obtain Mioduszewski's theorem on surjective universality of the pseudo-arc among chainable continua and a theorem on projective homogeneity of the pseudo-arc (which generalizes a result of Lewis and Smith on density of homeomorphisms of the pseudo-arc among surjective continuous maps from the pseudo-arc to itself). We also get a new characterization of the pseudo-arc via the projective homogeneity property.

Random continuum and Brownian motion

“…Assuming the claim, we show point (ii). Let K = K( B) be the continuum determined by B = (B n ) n∈N , as defined in (6). Assume…”

“…( (2) We recast the construction from point 1 above making it combinatorial. This is done by using the random walk, instead of the Brownian motion, and the point of view from [6]. We make a concrete choice for the sequence of random variables (T n ) and provide some detailed arguments.…”

“…and B(ω) determines a continuum} has measure 1. Recall definition(6), and consider the following function defined on Ω 0 :…”

European Congress of Mathematics Kraków, 2 – 7 July, 2012

“…However until very recently it remained open whether the countable atomless Boolean algebra admits ample generic automorphisms or equivalently whether the homeomorphism group of the Cantor space admits ample generics. This was finally resolved positively by Kwiatkowska [35], who used the powerful tools of the theory of dual or projective Fraïssé limits developed in Irwin-Solecki [26].…”

Stably projectionless Fraïssé limits

“…Many interesting objets in group theory, graph theory, and topology were identified as Fraïssé limits (see for example [17,Chapter 7] and [18]). Connections with Ramsey theory and topological dynamics, leading to the study of extreme amenability of the automorphism group of Fraïssé limits, were exploited in [21].…”