NEW YORK SEMINAR ON GENERAL TOPOLOGY AND TOPOLOGICAL ALGEBRA
September - December 2008
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DECEMBER 11: Ralph Kopperman, CCNY, CUNY, "Urysohn bases", at CCNY.
Tea 3pm, NAC 8/133, talk 4pm NAC 4/205. For more information, parking: R. Kopperman,
rdkcc@ccny.cuny.edu.
ABSTRACT: A Urysohn base is a simple notion of
neighborhood for sets, rather than points. It allows proof of the Urysohn lemma and the
so-called "sandwich" theorems that give conditions under which there is a continuous function
between a lower semicontinuous function and an upper semicontinuous function.
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NOVEMBER 13: Francis Jordan, Queensboro College, CUNY, "Function Space
Topologies", at CCNY. Tea 3pm, NAC 8/133, talk 4pm, NAC 4/205. For more Information, parking:
R. Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: In this talk we will consider when
various function space topologies coincide on the space of continuous function between spaces
X and Y. In particular, we will consider the compact-open, Isbell, and finest splitting
topologies. Another related problem is to find a concrete description of the finest splitting
topology. We will find such a description for the finest splitting topology on the upper
semicontinuous set-valued functions from X into Y. The restriction of this topology to the
continuous functions gives us what we call the fine Isbell topology. This new topology will
be used to distinguish between the standard function space topologies and answer a number of
open questions.
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NOVEMBER 20: Susan Andima, C. W. Post, LIU, "A Family of Asymmetric
Ellis-Type Theorems", at CCNY. Tea 3pm NAC 8/133, talk 4pm NAC 4/205. For more Information,
parking: R. Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: Amed Bouziad in 1996 generalized
famous theorems of Dean Montgomery (1936) and Robert Ellis (1957) to prove that every
Cech-complete space with a separately continuous group operation must be a topological group.
We generalize these and related theorems in a different direction by dropping the requirement
that the spaces be T1, so that our theorems apply to such "asymmetric" spaces as
the real numbers with the upper topology.
We develop an overall Ellis-type theorem for Hausdorff
k-bitoplogical spaces whose symmetrizations belong to a class of k-spaces with an Ellis-type
theorem. We apply our general result to particular properties and finally to some associated
asymmetric spaces with the use of a topological dual, \t^k, determined by a topology \t.
For example, the asymmetric theorem using properties of Cech-complete spaces is as follows.
Theorem. If (G,*,\t) is a group with a topology making
multiplication separately continuous and (G,\t,\t^k) is a Hausdorff k-bispace whose
symmetrization is Cech-complete, then multiplication is jointly continuous with respect to
both \t and \t^k, and inversion is a homeomorphism between (G,\t) and (G,\t^k).
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OCTOBER 16: Filiz Yildiz, Hacettepe University, Ankara, and CCNY,
"On Real Dicompact Extensions For Ditopological Spaces", at CCNY. Tea 3pm NAC 8/133,
talk 4pm NAC 4/205. For more Information, parking, contact: R. Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: Some important relationships are
obtained between the properties of a T-lattice (lattice with a suitable translation operation)
\BA(S) of w-preserving, bicontinuous, real-valued functions and the ditopological
properties of the ditopological texture space S.
By using these relations, a generalization of classical
realcompactness is defined for ditopological spaces under the name real dicompactness.
In this talk, a notion of real dicompact extension is defined for
ditopological spaces and characterized for those ditopological spaces that posses a real
dicompact extension.
Finally,a suitable counterpart of the classical Hewitt
Realcompactification is obtained for ditopological spaces in texture theory.
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OCTOBER 23: Stephen A. Saxon, University of Florida, "Mackey hyperplanes
and enlargements", at Baruch College Vertical Campus, Lexington Avenue and 24th Street.
Tea 3:15, talk 4 p.m., Mathematics Conference Room, 6-215. For more information, contact A. Todd,
artodd@panix.com.
ABSTRACT: Taqdir Husain (1966) articulated the
Aleph0-barrelled concept amid nondistinguished Fr‚chet spaces characterized by
Aleph0-barrelled strong duals that are not barrelled. When Saxon/Tweddle (1999)
optimally extended certain theorems from barrelled spaces to Mackey Aleph0-barrelled
spaces, they gave the (only known?) example of a Mackey Aleph0-barrelled space that
is not barrelled. Tweddle's invention (linfty,eta), with its topology eta defined
via a mad family on N, fell short: It is far from Aleph0-barrelled. Nevertheless,
Tweddle (2008) showed it has interesting properties and is, indeed, Mackey. Some Mackey spaces
admit nonMackey hyperplanes [Levin/Saxon, Tweddle/Saxon, Valdivia]. Does (linfty,eta)?
This paper will twice answer a robust No: subspaces of codimension < c are Mackey, as are
countable enlargements. A (harder) proof of the same for the Saxon/Tweddle example will generate
more known spaces of its type.
Tweddle kindly credits Mackeyness of (linfty,eta) to
Saxon's 1997 personal reply. The more robust answers mentioned use a Todd/Saxon strong
barrelledness theorem (1973) and occasion new Mackey three-space/enlargement theorems that
enhance the foundations of Valdivia, Tweddle/Saxon, Robertson/Tweddle/Yeomans, and De Wilde/
Tsrulnikov, for whom novel and simpler proofs are given.
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OCTOBER 30: October 30: Aaron Todd, Baruch College, "The strongly Hewitt
property of X and strong barrelled/weak Baire properties for Cc(X)", at CCNY.
Tea 3pm NAC 8/133, talk 4pm, NAC 4/205. For more Information, parking, contact: R. Kopperman,
rdkcc@ccny.cuny.edu.
ABSTRACT: For X a Tychonov space and
Cc(X) the locally convex space of continuous realvalued functions on X under
the compact-open topology, we have answered a question related to the strongly Hewitt property
for X defined by Kakol and Sliwa and the unordered Bairelike property on Cc(X)
defined by Saxon. Originally raised in a paper of Todd and Render, this question was again
raised by the speaker at a mini-conference on problems sponsored by our seminar this past
summer. Some details of this answer are discussed with the aim of placing it in the larger
context of interactions between topological properties of X and functional analytic properties
of Cc(X).
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SEPTEMBER 4: Steve Matthews, U. of Warwick, UK, "Partial metric spaces",
at CCNY. Tea 3pm NAC 8/133, talk 4pm NAC 4/205. For more information, parking, contact:
R. Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: Introduced in 1992, a 'partial
metric space' is a generalization of the notion of 'metric space' defined in 1906 by
Maurice Frechet such that the distance of a point from itself is not necessarily zero.
Motivated by needs in computer science for non Hausdorff Scott topology, we show that much
of the essential structure of metric spaces, such as Banach's contraction mapping theorem,
can be generalized to allow for the possibility of non zero self-distances d(x,x).
This talk will introduce the essential motivation, theory, and applications for partial
metric spaces, leading to the conclusion that the non Hausdorff nature of topology in
computer science is calling upon metric topology to reconsider its foundations.
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SEPTEMBER 11: Ralph Kopperman, CCNY, "Spherical completion of partial
metric spaces and some uses", at CCNY. Tea 3pm NAC 8/133, talk 4pm 4/205. For more Information,
parking, contact: R. Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: Topologies arising from partial
metrics are not Hausdorff, and usually have nontrivial specialization orders (where
x\leq y means x\in\cl(y)). A partial metric space is complete if each Cauchy net
has a limit in the topology arising from the associated metric (defined by dp(x,y)=
2p(x,y)-p(x,x)-p(x,y)). It is spherically complete if Cauchy nets that are order preserving
with respect to the specialization has such a limit. Each partial metric space has a (unique)
completion and spherical completion.
We discuss two results: The round ideal completion is a
generalization of the Dedekind order completion that gives the reals from the the rationals.
In domain theory it is used to get continuous posets. We show how to obtain the round ideal
completion as a spherical completion.
Quantale-valued partial metrics have been used to give a
foundation to the theory of fuzzy sets. We discuss a model associated with his work and show
that a spherically complete such model satisfies the axiom of choice.
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SEPTEMBER 4: Michael Bukatin, MetaCarta Inc., Cambridge, MA:
"Distance and Equality for Partially Defined Elements", at CCNY. Tea 3pm NAC 8/133,
talk 4pm TBA. For more Information, parking, contact: R. Kopperman,
rdkcc@ccny.cuny.edu.
ABSTRACT: One approach to a theory of
partial elements is the theory of partial orders with the Scott topology. Distances
without the "p(x,x)=0" axiom represent the computationally correct way to produce
generalized metrizations of this non-Hausdorff topology. Studies of distances without the
"p(x,x)=0" axiom led to the theory of partial metrics, and, eventually, to that of
quantale-valued partial metrics.
Another approach to the theory of partial elements is the
theory of sheaves and presheaves. A generalized equality appears in the context of presheaves:
for example, the degree of equality of two partially defined functions is often defined as the
domain on which their values coincide. Then a partial function equals to itself only to the
extent to which it is defined. So we have an equality without the "=(x,x) = true" axiom. This
equality is generally valued not in "{false,true}", but in a powerset, a topology, or some
other Boolean or Heyting algebra, but otherwise it resembles a partial ultrametric.
Ulrich Hoehle further generalized this equality to a
quantale-valued fuzzy equality in the process of giving a solid foundation to the theory of
fuzzy sets (and, in particular, to the use of Lukasiewicz logic).
It turns out that quantale-valued partial metrics and
quantale-valued fuzzy equalities coincide. We'll explore some implications of this.
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For more information, contact:
CCNY (212-650-5346): R. Kopperman
College of Staten Island
(718-982-3626): P. R. Misra
Baruch College (212-387-1463): A. Todd
LIU C.
W. Post Center (516-299-2447): S. Andima
Queens College, Math. (718-380-1832):
G. Itzkowitz
Queens College, Comp. Sci. (718-997-3478): T. Y. Kong
Medgar Evers
(718-270-6125): H. Pajoohesh