Compact spaces generated by retractions

We study compact spaces which are obtained from metric compacta by iterating the operation of inverse limit of continuous sequences of retractions. We denote this class by R. Allowing continuous images in the definition of class R, one obtains a strictly larger class, which we denote by RC. We show that every space in class RC is either Corson compact or else contains a copy of the ordinal segment $[0,\omega_1]$. This improves a result of Kalenda, where the same was proved for the class of continuous images of Valdivia compacta. We prove that spaces in class R do not contain cutting P-points (see the definition below), which provides a tool for finding spaces in RC minus R. Finally, we study linearly ordered spaces in class RC. We prove that scattered linearly ordered compacta belong to RC and we characterize those ones which belong to R. We show that there are only 5 types (up to order isomorphism) of connected linearly ordered spaces in class R and all of them are Valdivia compact. Finally, we find a universal pre-image for the class of all linearly ordered Valdivia compacta.Comment: Minor corrections; added two statements on linearly ordered compacta. The paper has 21 pages and 2 diagram

A Universal Continuum of Weight aleph

We prove that every continuum of weight aleph_1 is a continuous image of the Cech-Stone-remainder R^* of the real line. It follows that under CH the remainder of the half line [0,infty) is universal among the continua of weight c --- universal in the `mapping onto' sense. We complement this result by showing that 1) under MA every continuum of weight less than c is a continuous image of R^* 2) in the Cohen model the long segment of length omega_2+1 is not a continuous image of R^*, and 3) PFA implies that I_u is not a continuous image of R^*, whenever u is a c-saturated ultrafilter. We also show that a universal continuum can be gotten from a c-saturated ultrafilter on omega and that it is consistent that there is no universal continuum of weight c.Comment: 15 pages; 1999-01-27: revision, following referee's report; improved presentation some additional results; 2000-01-24: final version, to appear in Trans. Amer. Math. So

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Davies-trees in infinite combinatorics

This short note, prepared for the Logic Colloquium 2014, provides an introduction to Davies-trees and presents new applications in infinite combinatorics. In particular, we give new and simple proofs to the following theorems of P. Komj\'ath: every $n$-almost disjoint family of sets is essentially disjoint for any $n\in \mathbb N$; $\mathbb R^2$ is the union of $n+2$ clouds if the continuum is at most $\aleph_n$ for any $n\in \mathbb N$; every uncountably chromatic graph contains $n$-connected uncountably chromatic subgraphs for every $n\in \mathbb N$.Comment: 8 pages, prepared for the Logic Colloquium 201

The Chang-Los-Suszko Theorem in a Topological Setting

The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications

An Algebraic and Logical approach to continuous images

Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these continuous mappings -- also tools from Model Theory. We illustrate by showing that the \v{C}ech-Stone remainder $[0,\infty)$ has a universality property akin to that of $N^*$; a theorem of Ma\'ckowiak and Tymchatyn implies it own generalization to non-metric continua; and certain concrete compact spaces need not be continuous images of $N^*$.Comment: Notes from a series of lectures at http://www.cts.cuni.cz/events/ws/2002/ws2002.htm, the 30th Winter School on Abstract Analysis 2002-05-02: corrected version after referee's repor