Work in Progress

March 2012

Heike Mildenberger and Dilip Raghavan and J. Steprans

A question of the second two authors is answered by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost disjoint family under $\mathfrak{s} \leq \mathfrak{a}$, partially answering a question of Shelah.

August 2011

S. Shelah and J. Steprans

Nontrivial automorphisms of the $P(\mathbb{N})/[\mathbb{N}]^{<\aleph_0}$ are constructed using that a variant of the dominating number is small.

August 2011

J. Steprans

Under Martin's Axiom the $w^*$-closure of fewer than continuum singular elements of $L_1^{**}$ is contained in the set of singular elements of $L_1^{**}$.

July 2011

Samuel Coskey, Tamas Matrai, and Juris Steprans

The Borel Tukey ordering on cardinal invariants of the continuum is considered and it is observed that this ordering makes sense for a larger class of cardinals than has previously been considered. A Borel version of a large portion of the van Douwen diagram is provided. For instance, although the usual proof of the inequality $\mathfrak{p}\leq \mathfrak{b}$ does not provide a Borel Tukey map, in fact there is one.

June 2011

D. Raghavan and J. Steprans

The amenability character of a Banach algebra is defined and some elementary observations about it are made. In particular, it is shown that $C(X)$ has countable amenability character if and only if $X$ is metrizable.

June 2011

D. Raghavan and J. Steprans

This is a start at showing there is a Michael space in the Laver model.

May 2011

J. Steprans

A question of V. Pestov on amenable actions is answered using a construction of M. Foreman.

May 2011

J. Steprans

These are slides of a four lecture series on amenability presented at the Young Set Theorists meeting in Bonn in 2011. The last lecture contains joint work with D.Raghavan extending Foreman's group construction

July 2010

K. Kunen, J. Larson and J. Steprans
arXiv:1008.4739

Assuming $\diamond$: Whenever $B$ is a totally imperfect set of real numbers, there is special Aronszajn tree with no continuous order preserving map into $B$.

June 2010

V. Losert, M. Neufang, J. Pachl and J. Steprans

The Ghahramani--Lau conjecture is established; in other words, the measure algebra of every locally compact group is strongly Arens irregular.

June 2010

D. Raghavan and J. Steprans

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $2^{\aleph_0} <\aleph_\omega$ a weakly tight family is constructed under the hypothesis $\mathfrak{s} < \mathfrak{b} < \aleph_\omega$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrusak and Garcia Ferreira, who applied it to the Katetov order on almost disjoint families.

December 2010

M. Elekes and J. Steprans

It is consistent that every non meagre subset of the Erdos-Kakutani group intersects a translate of a small set on a relatively non meagre set.

November 2010

P. Larson, A. Miller, J. Steprans and W. Weiss

A question from the Scottish Book on universal functions is considered.

Articles to Appear

In Press 1

Maximal Functions and the Additivity of Various Families of Null Sets
Juris Steprans
to appear in TAMS    (2010)

In Press 2

History of the Continuum in the Twentieth Century
Juris Steprans
to appear as part of Volume 6 of the History of Logic    (2011)

In Press 3

Masas in the Calkin Algebra without the Continuum Hypothesis
Saharon Shelah and Juris Steprans
to appear in Fund. Math.    (2010)

Reviewed Articles

MR2658145

The commutant of L(H) in its ultrapower may or may not be trivial
Ilijas Farah, N. Christopher Phillips, and Juris Steprans
Mathematische Annalen  347  839--857  (2010)
DOI: 10.1007/s00208-009-0448-z

MR2467209

Analytic and coanalytic families of almost disjoint functions
B. Kastermans and J. Steprans and Y. Zhang
J. Symbolic Logic  73  1158--1172  (2008)
http://projecteuclid.org/getRecord?id=euclid.jsl/1230396911

MR2462464

Chasing Silver
A. Roslanowski and J. Steprans
Canad. Math. Bull.  51  593--603  (2008)
DOI: 10.4153/CMB-2008-059-2

MR2457482

The consistency of ${\mathfrak b}=\kappa$ and ${\mathfrak s} =\kappa^{+}$
V. Fischer and J. Steprans
Fund. Math.  201  283--293  (2008)
DOI: 10.4064/fm201-3-5

MR2353856

Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers
S. Shelah and J. Steprans
Fund. Math.  196  197--235  (2007)
DOI: 10.4064/fm196-3-1

MR2258628

Products of sequential CLP-compact spaces are CLP-compact
J. Steprans
Ann. Pure Appl. Logic  143  155--157  (2006)
DOI: 10.1016/j.apal.2005.03.005

MR2224048

The number of translates of a closed nowhere dense set required to cover a Polish group
A. W. Miller and J. Steprans
Ann. Pure Appl. Logic  140  52--59  (2006)
DOI: 10.1016/j.apal.2005.09.010

MR2214123

Chains of Baire class 1 functions and various notions of special trees
M. Elekes and J. Steprans
Israel J. Math.  151  179--187  (2006)
DOI: 10.1007/BF02777361

MR2198711

Geometric cardinal invariants, maximal functions and a measure theoretic pigeonhole principle
J. Steprans
Bull. Symbolic Logic  11  517--525  (2005)
http://projecteuclid.org/getRecord?id=euclid.bsl/1130335207

MR2167583

Many quotient algebras of the integers modulo co-analytic ideals
J. Steprans
Contemp. Math. Logic and its applications  380  271--281  (2005)

MR2128705

Comparing the uniformity invariants of null sets for different measures
S. Shelah and J. Steprans
Adv. Math.  192  403--426  (2005)
DOI: 10.1016/j.aim.2004.04.010

MR2092071

Complete Erdos space is unstable
J. J. Dijkstra and J. Van Mill and J. Steprans
Math. Proc. Cambridge Philos. Soc.  137  465--473  (2004)
DOI: 10.1017/S0305004104007996

MR2071696

Less than $2^\omega$ many translates of a compact nullset may cover the real line
M. Elekes and J. Steprans
Fund. Math.  181  89--96  (2004)
DOI: 10.4064/fm181-1-4

MR1990584

The autohomeomorphism group of the Cech-Stone compactification of the integers
J. Steprans
Trans. Amer. Math. Soc.  355  4223--4240 (electronic)  (2003)
DOI: 10.1090/S0002-9947-03-03329-4

MR1896046

Martin's axiom is consistent with the existence of nowhere trivial automorphisms
S. Shelah and J. Steprans
Proc. Amer. Math. Soc.  130  2097--2106 (electronic)  (2002)
DOI: 10.1090/S0002-9939-01-06280-3

MR2032852

The uniformity invariants of the ideal of null sets in separable metric spaces
J. Steprans
Topology Proc.  26  811--818  (2001/02)

MR1923169

The almost disjointness cardinal invariant in the quotient algebra of the rationals modulo the nowhere dense subsets
J. Steprans
Real Anal. Exchange  27  795--800  (2001/02)

MR1856740

Cofinitary groups, almost disjoint and dominating families
M. Hrusak and J. Steprans and Y. Zhang
J. Symbolic Logic  66  1259--1276  (2001)
DOI: 10.2307/2695105

MR1855551

Cardinal invariants related to sequential separability
M. Hrusak and J. Steprans
S\=urikaisekikenky\=usho K\=oky\=uroku    66--74  (2001)

MR1833473

The covering numbers of Mycielski ideals are all equal
S. Shelah and J. Steprans
J. Symbolic Logic  66  707--718  (2001)
DOI: 10.2307/2695039

MR1733805

Restricted compactness properties and their preservation under products
J. Steprans and A. P. Sostak
Topology Appl.  101  213--229  (2000)
DOI: 10.1016/S0166-8641(98)00126-6

MR1777780

Unions of rectifiable curves in Euclidean space and the covering number of the meagre ideal
J. Steprans
J. Symbolic Logic  64  701--726  (1999)
DOI: 10.2307/2586494

MR1473455

Decomposing Euclidean space with a small number of smooth sets
J. Steprans
Trans. Amer. Math. Soc.  351  1461--1480  (1999)
DOI: 10.1090/S0002-9947-99-02197-2

MR1610762

Orthogonal families of real sequences
A. W. Miller and J. Steprans
J. Symbolic Logic  63  29--49  (1998)
DOI: 10.2307/2586584

MR1625478

Maximal filters, continuity and choice principles
H. Herrlich and J. Steprans
Quaestiones Math.  20  697--705  (1997)

MR1450995

$G_\delta$-sets in topological spaces and games
W. Just and M. Scheepers and J. Steprans and P. J. Szeptycki
Fund. Math.  153  41--58  (1997)

MR1433612

Hausdorff capacity and Lebesgue measure
T. S. Salisbury and J. Steprans
Real Anal. Exchange  22  265--278  (1996/97)

MR1405489

Reaping numbers of Boolean algebras
A. Dow and J. Steprans and S. Watson
Bull. London Math. Soc.  28  591--599  (1996)
DOI: 10.1112/blms/28.6.591

MR1367140

Cardinal invariants associated with Hausdorff capacities
J. Steprans
  192  147--184  (1996)

MR1314977

Sums of Darboux and continuous functions
J. Steprans
Fund. Math.  146  107--120  (1995)

MR1301530

Mutually complementary families of $T_1$ topologies, equivalence relations and partial orders
J. Steprans and S. Watson
Proc. Amer. Math. Soc.  123  2237--2249  (1995)
DOI: 10.2307/2160963

MR1303686

A topological Banach fixed point theorem for compact Hausdorff spaces
J. Steprans and S. Watson and W. Just
Canad. Math. Bull.  37  552--555  (1994)

MR1297403

Decomposing Baire class 1 functions into continuous functions
S. Shelah and J. Steprans
Fund. Math.  145  171--180  (1994)

MR1271551

Somewhere trivial autohomeomorphisms
S. Shelah and J. Steprans
J. London Math. Soc. (2)  49  569--580  (1994)

MR1271434

Erratum: ``Maximal chains in ${}^\omega\omega$ and ultrapowers of the integers'' [Arch. Math. Logic {\bf 32} (1993), no.\ 5, 305--319; {MR}1223393 (94g:03094)]
S. Shelah and J. Steprans
Arch. Math. Logic  33  167--168  (1994)
DOI: 10.1007/BF01352936

MR1261556

The $\sigma$-linkedness of the measure algebra
A. Dow and J. Steprans
Canad. Math. Bull.  37  42--45  (1994)

MR1259349

Homogeneous almost disjoint families
S. Shelah and J. Steprans
Algebra Universalis  31  196--203  (1994)
DOI: 10.1007/BF01236517

MR1253008

Dominating functions and graphs
R. Diestel and S. Shelah and J. Steprans
J. London Math. Soc. (2)  49  16--24  (1994)

MR1252708

The evolution of integration
A. Shenitzer and J. Steprans
Amer. Math. Monthly  101  66--72  (1994)
DOI: 10.2307/2325128

MR1253921

A very discontinuous Borel function
J. Steprans
J. Symbolic Logic  58  1268--1283  (1993)
DOI: 10.2307/2275142

MR1239060

Martin's axiom and the transitivity of $P_{\mathfrak c}$-points
J. Steprans
Israel J. Math.  83  257--274  (1993)
DOI: 10.1007/BF02784054

MR1234290

Combinatorial consequences of adding Cohen reals
J. Steprans
  6  583--617  (1993)

MR1229497

Continuous colourings of closed graphs
A. Krawczyk and J. Steprans
Topology Appl.  51  13--26  (1993)
DOI: 10.1016/0166-8641(93)90011-2

MR1223393

Maximal chains in ${}^\omega\omega$ and ultrapowers of the integers
S. Shelah and J. Steprans
Arch. Math. Logic  32  305--319  (1993)
DOI: 10.1007/BF01409965

MR1217174

Combinatorial properties of the ideal $P_2$
J. Cichon and A. Roslanowski and J. Steprans and B. Weglorz
J. Symbolic Logic  58  42--54  (1993)
DOI: 10.2307/2275322

MR1186465

Countable Frechet $\alpha_1$-spaces may be first countable
A. Dow and J. Steprans
Arch. Math. Logic  32  33--50  (1992)
DOI: 10.1007/BF01270393

MR1168663

Topological invariants in the Cohen model
J. Steprans
Topology Appl.  45  85--101  (1992)
DOI: 10.1016/0166-8641(92)90050-A

MR1206463

Almost disjoint families of paths in lattice grids
J. Steprans
Topology Proc.  16  185--200  (1991)

MR1078638

Steprans' problems
J. Steprans
    13--20  (1990)

MR1002627

Nontrivial homeomorphisms of $\beta N\setminus N$ without the continuum hypothesis
S. Shelah and J. Steprans
Fund. Math.  132  135--141  (1989)

MR985683

A model in which countable Frechet $\alpha_1$-spaces are first countable
A. Dow and J. Steprans
Math. Proc. Cambridge Philos. Soc.  105  473--480  (1989)

MR962545

Destroying precaliber $ \aleph_1$: an application of a -system lemma for closed sets
J. Steprans and S. Watson
Fund. Math.  129  223--229  (1988)

MR958051

A Banach space on which there are few operators
S. Shelah and J. Steprans
Proc. Amer. Math. Soc.  104  101--105  (1988)
DOI: 10.2307/2047469

MR935111

PFA implies all automorphisms are trivial
S. Shelah and J. Steprans
Proc. Amer. Math. Soc.  104  1220--1225  (1988)
DOI: 10.2307/2047617

MR932979

Some results on CDH spaces. I
J. Steprans and H. X. Zhou
Topology Appl.  28  147--154  (1988)
DOI: 10.1016/0166-8641(88)90006-5

MR932978

Cellularity of first countable spaces
J. Steprans and S. Watson
Topology Appl.  28  141--145  (1988)
DOI: 10.1016/0166-8641(88)90005-3

MR887767

Homeomorphisms of manifolds with prescribed behaviour on large dense sets
J. Steprans and W. S. Watson
Bull. London Math. Soc.  19  305--310  (1987)
DOI: 10.1112/blms/19.4.305

MR887554

Extraspecial p-groups
S. Shelah and J. Steprans
Ann. Pure Appl. Logic  34  87--97  (1987)
DOI: 10.1016/0168-0072(87)90041-8

MR801424

Strong Q-sequences and variations on Martin's axiom
J. Steprans
Canad. J. Math.  37  730--746  (1985)

MR770551

A characterization of free abelian groups
J. Steprans
Proc. Amer. Math. Soc.  93  347--349  (1985)
DOI: 10.2307/2044776

MR769852

The number of directed sets
K. J. Devlin and J. Steprans and W. S. Watson
Rend. Circ. Mat. Palermo (2)    31--41  (1984)

MR743378

The number of submodules
J. Steprans
Proc. London Math. Soc. (3)  49  183--192  (1984)
DOI: 10.1112/plms/s3-49.1.183

MR1540295

Problems and Solutions: Solutions of Advanced Problems: 6380
J. Steprans
Amer. Math. Monthly  90  649  (1983)
http://links.jstor.org/sici?sici=0002-9890(198311)90:9<649:6>2.0.CO;2-K&origin=MSN

MR1539901

Problems and Solutions: Advanced Problems: 6380-6382
J. Steprans and W. W. Meyer and H. Lam
Amer. Math. Monthly  89  214  (1982)
DOI: 10.2307/2320210

MR633292

Cardinal arithmetic and $\aleph _{1}$-Borel sets
J. Steprans
Proc. Amer. Math. Soc.  84  121--126  (1982)
DOI: 10.2307/2043823

MR612014

Trees and continuous mappings into the real line
J. Steprans Žurnāls
Topology Appl.  12  181--185  (1981)
DOI: 10.1016/0166-8641(81)90019-5