If $U$ is a normal ultrafilter on a measurable cardinal $\kappa$, then the intersection of the $\omega$ first iterated ultrapowers of the universe by $U$ is a Prikry generic extension of the $\omega$th iterated ultrapower.

Here, we analyse some recent applications of set theory to topology and argue that set theory is not only the closed domain where mathematics is usually founded, but also a flexible framework where imperfect intuitions can be precisely formalized and technically elaborated before they possibly migrate toward other branches. This apparently new role is mostly reminiscent of the one played by other external fields like theoretical physics, and we think that it could contribute to revitalize the interest in set theory (...) in the future. (shrink)

If $U_\alpha$ is a length $\omega_1$ sequence of normal ultrafilters on a measurable cardinal $\kappa$ that is increaing w.r.t. the Mitchel order, then the intersection of the $\omega_1$ first iterated ultrapowers of the universe is a Magidor generic extension of the $\omega_1$th iterated ultrapower.

Associate to any linear ordering on the integers the mapping whose value on n is the cardinality of {kn; kn}: a purely combinatorial characterization for the mappings associated to the well-orderings is established.

In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimen\-sional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.