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# Santi Spadaro

Santi Spadaro

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

### Project information

#### Title

Cardinal Functions and Discrete Sets

#### Outgoing host organisation Return host organisation

York University Universita degli Studi di Catania

Department of Mathematics and Statistics Dipartimento di Matematica e Informatica

4700 Keele Street Viale Andrea Doria, 6

M3J 1P3 Toronto, Canada 98125 Catania, Italy

http://www.yorku.ca/ http://www.dmi.unict.it/

#### 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

- S. Spadaro,
*Covering by discrete and closed*, Topology and its Applications, 156 (2009), 721-727

discrete sets - S. Spadaro,
*A note on discrete*, Commentationes Mathematicae Universitatis Carolinae, 50,

sets

3 (2009), 463-475 - A. Bella, C. Costantini, S. Spadaro,
*P-spaces and*, Houston Journal of Mathematics, 37, 3 (2011),

the Whyburn Property

995-1015 - M. Kojman, D. Milovich,

S. Spadaro,*Order-theoretic properties of bases in topological*, submitted, 20 pages

spaces, I - A. Bella, M. Matveev, S. Spadaro,
*Variations of*, Topology and its Applications, 159,

selective separability, II: discrete sets and the influence of

convergence and maximality

1 (2012), 253-271 - S. Spadaro,
*A short proof of a theorem of*, Topology and its Applications, 158 (2011), 2091-2093

JuhÃ¡sz - S. Spadaro,
*On two cardinal invariants of an*, Annals of Pure and Applied Logic,

order-theoretic flavour

163 (2012), 1865-1871 - S. Spadaro,
*P-spaces and the Volterra*, Bulletin of the Australian Mathematical Society, in press

Property - D. Soukup, L.Soukup, S. Spadaro,
*Comparing weak versions of separability*, preprint (2012); arXiv:1210.4986, 29 pages, submitted - A. Bella, S. Spadaro,
*Infinite games and cardinal properties of topological spaces*, preprint (2012); arXiv:1212.5725, 14 pages, submitted - N. May, S. Spadaro, P. Szeptycki,
*A new class of*, 20 pages,

Lindelof spaces all of whose finite powers are Lindelof

submitted (arXiv preprint to come soon) - S. Spadaro,
*Discrete sets and cardinal*, in preparation

invariants

#### 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

UniversitÃ 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

UniversitÃ 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