Partition calculus for topological spaces

grantor: University of TorontoThe subject matter of this exposition is the study of partition relations for particular topological spaces. For topological spaces X and Y, and for a positive integer n and a cardinal k the notation $X\longrightarrow(Y)\sbsp{k}{n}$ means that for any partition f of $\lbrack X\rbrack\sp{n}$ into k pieces, there is a subspace H of X homeomorphic to Y and f is constant on $\lbrack H\rbrack\sp{n}.$ In contrast with the ordinary partition calculus in this area very few positive results are known. For example, it is consistent that for $n > 3$ and for any space X, we have $X\not\longrightarrow(Y)\sbsp{2}{n}$ for every topological spaces Y in which not all the points are isolated. Even in the case of $n = 3$ if the space Y is complicated enough the relation fails to be positive. It seems that the cases $n = 1$ and 2 are the cases where there exists some positive relations for non-trivial space Y. The spaces considered in this exposition are the followings: Ordinal spaces, Reals, Rationals, Cantor Cubes, $\Sigma$-products, Cech-Stone compactification of the integers, Alexandroff compactification of discrete spaces and the countable sequences converging to a point in the sense of an arbitrary ultrafilter of ${\rm I\!N},$ denoted by $\omega\cup\{p\}.$ We will present and review many techniques for proving topological relations when $n = 1.$ Then we shall use a stepping-up procedure to obtain results for higher dimensions $n > 1$ when possible. Using ordinary axioms of set theory (ZFC), we will establish some positive relations for the space $2\sp\kappa$ such as $2\sp\kappa\longrightarrow(2\sp\kappa, \doubq)\sp1,\ 2\sp{c}\longrightarrow(\omega\cup\{p\})\sbsp{\omega}{1}$ and $\Sigma\sb{\aleph\sb2}\not\longrightarrow(\aleph\sb2)\sbsp{3}{1},$ where $\Sigma\sb{\aleph\sb2}$ denotes $\Sigma\sb{\aleph\sb2}\{0,1\},$ the sigma-product of $\aleph\sb2$ copies of the two point discrete space $\{0,1\}.$ Beyond ZFC, we introduce and apply the method of elementary submodels and use the SPFA and some of its consequences to establish the consistency of $2\sp{\omega\sb1}\longrightarrow(\omega\sb1 + 1, B(\omega\sb1))\sp1$ as well as the consistency of $2\sp{\omega\sb1}\longrightarrow(\omega\sb1 + 1,\omega\sb1)\sp1.$ The additional axiom SPFA has a considerable large cardinal consistency strength, but we shall establish that the large cardinal assumption is necessary for the consistency of the positive relation $\Sigma\sb{\aleph\sb2}\longrightarrow(\omega\sb1)\sbsp{2}{1}.$ The techniques of forcing and iterated forcing are used to define generic partitions for some spaces leading to the independence of the partition of the form $\Sigma\sb{\aleph\sb2}\longrightarrow(\omega\sb1)\sbsp{2}{1}.$Ph.D