Santi Spadaro

INdAM-COFUND Outgoing fellow between 2012-02-01 and 2013-01-31

Project information

Title

Cardinal Functions and Discrete Sets

Abstract

My project will investigate discrete sets in connection with covering properties (D-spaces), cardinal functions (discrete reflection of cardinality) and selection principles (D-separability). It will also investigate a new class of cardinal functions of an order-theoretic flavor which have been shown to be amenable to the techniques of Shelah’s PCF theory and sensitive
to very large cardinals.

In the first part I plan to attack some problems regarding chain-condition-type properties in products, in particular to study the influence of Chang Conjecture and forcing axioms on their behavior. In the second part I plan to study some properties involving discrete sets and nowhere dense sets similar to properties in the framework of Selection Principles in Mathematics and try to isolate new selection principles inspired by these properties. In the third part I plan to study the way that closures of discrete sets reflect properties of topological spaces (especially cardinality and covering properties). In particular I will try to give more partial positive answers to Arhangel’skii’s question asking whether closures of discrete sets reflect cardinality in compact spaces. Moreover, I plan to attack some special cases of van Douwen’s D-space problem, for example, the construction of a T1 counterexample from the usual axioms of Zermelo-Fraenkel Set Theory.

Fellow information

Previous positions, awards

• August 2011-February 2012: Visiting Assistant Professor, Miami University,
  USA
• 2011: INFTY Short Visit Grant, European Mathematical
  Society
• 2010-2011: VATAT grant, Israeli Council of Higher
  Education
• 2009-2011: Postdoctoral Fellow, Center for Advanced Study,
  Ben Gurion University of the Negev, Israel
• 2008: Ben Fitzpatrick Memorial Fellowship, Auburn
  University

Publications, preprints, other works

1. S. Spadaro, Covering by discrete and closed
   discrete sets, Topology and its Applications, 156 (2009), 721-727
2. S. Spadaro, A note on discrete
   sets, Commentationes Mathematicae Universitatis Carolinae, 50,
   3 (2009), 463-475
3. A. Bella, C. Costantini, S. Spadaro, P-spaces and
   the Whyburn Property, Houston Journal of Mathematics, 37, 3 (2011),
   995-1015
4. M. Kojman, D. Milovich,
   S. Spadaro, Order-theoretic properties of bases in topological
   spaces, I, submitted, 20 pages
5. A. Bella, M. Matveev, S. Spadaro, Variations of
   selective separability, II: discrete sets and the influence of
   convergence and maximality, Topology and its Applications, 159,
   1 (2012), 253-271
6. S. Spadaro, A short proof of a theorem of
   Juhász, Topology and its Applications, 158 (2011), 2091-2093
7. S. Spadaro, On two cardinal invariants of an
   order-theoretic flavour, Annals of Pure and Applied Logic,
   163 (2012), 1865-1871
8. S. Spadaro, P-spaces and the Volterra
   Property, Bulletin of the Australian Mathematical Society, in press
9. D. Soukup, L.Soukup, S. Spadaro, Comparing weak versions of separability, preprint (2012); arXiv:1210.4986, 29 pages, submitted
10. A. Bella, S. Spadaro, Infinite games and cardinal properties of topological spaces, preprint (2012); arXiv:1212.5725, 14 pages, submitted
11. N. May, S. Spadaro, P. Szeptycki, A new class of
    Lindelof spaces all of whose finite powers are Lindelof, 20 pages,
    submitted (arXiv preprint to come soon)
12. S. Spadaro, Discrete sets and cardinal
    invariants, in preparation

Conferences, schools, other events

• Invited talk (title to be announced)
  Logic Seminar
  University of Colorado-Boulder, USA, 16 April 2013
• Invited talk (title to be announced)
  at the Special Session in Boolean Algebras
  and Set Theory, Spring Western Sectional Meeting of the American
  Mathematical Society
  Boulder, Colorado, USA, 13-14 April 2013
• Short talk Introduction to Elementary Submodels
  at the Eighth Italian-Spanish Conference on
  General Topology and its Applications
  Università degli Studi di Trieste, Trieste, Italy, 3-7 September 2012
• One-hour talk Noetherian type of products
  at the Eighth Italian-Spanish Conference on
  General Topology and its Applications
  Università degli Studi di Trieste, Trieste, Italy, 3-7 September 2012
• Job talk Independence results about an order-theoretic cardinal invariant
  at the University of Wisconsin-Parkside
  Kenosha, WI, USA, 5 March 2012
• Talk Noetherian type and other topological
  cardinal invariants of an order-theoretic flavour
  at the Set Theory Seminar, Fields Institute, Toronto
  Ontario, Canada, 2 March 2012