NEW YORK SEMINAR ON GENERAL TOPOLOGY AND TOPOLOGICAL ALGEBRA

February - May 2012

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Go to March 8 talk    Go to March 22 talk    Go to April 5 talk    Go to April 19 talk    Go to May 3 talk    Go to May 31 talk

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FEBRUARY 16:   Homeira Pajoohesh Jordan, Medgar Evers College, CUNY, "K-metric Spaces", Long Island University - C. W. Post. Tea at 3:15 p.m. by the Math Department office, Pell Hall room 240; talk at 4:00 p.m. in Pell Hall room 202. For local information, contact S. Andima, sandima@liu.edu.

     ABSTRACT:   In this talk we give a new generalization of metric spaces called k-metric spaces. Our k-metrics are valued in lattice ordered groups, which allows us to talk about distance in non-abelian lattice ordered groups. We also discuss a class of (not necessarily abelian) lattice ordered groups in which every k-metric induces a topology. Then we show that every k-metric valued in the real numbers is metrizable.


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MARCH 8:   Francis Jordan, Queensboro College, CUNY, "When the finest splitting topology on C(X) is a group topology", at CCNY. Tea 3pm NAC 8/133, talk 4pm NAC 4/113. For parking, more information, contact: R. Kopperman, rdkcc@ccny.cuny.edu.

     ABSTRACT:   We characterize the completely regular Lindelof spaces X for which C(X) with the finest splitting topology is a topological group as those spaces that are infraconsonant and sequentially inaccessible. Being sequentially inaccessible is also shown to be equivalent to C(X) with the finest splitting topology being Frechet when X is completely regular and Lindelof. We will also discuss the relationship of this work with the old problem of finding (in ZFC) a separable Frechet topological group that is not metrizable.

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MARCH 22:   Ralph Kopperman, City College, CUNY, "How categories of closure spaces arise from generalized metrics", at CCNY. Tea 3pm, NAC 8/133, talk 4pm, NAC 4/205. For more information, parking: R. Kopperman, rdkcc@ccny.cuny.edu.

     ABSTRACT:   We recall how topological spaces arise from generalized metrics, and show how to obtain more general categories of closure spaces similarly.

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APRIL 5:   Ivan S. Gotchev (Note (1)), Central Connecticut State University, "Continuous extensions of functions defined on subsets of products with the \kappa-box topology". At Baruch College, CUNY, Vertical Campus, Lexington Avenue and 24th Street. Tea at 3:15 p.m., talk at 4:00 p.m., both in Room VC 6-215. For local information: A. Todd, aaron.todd@baruch.cuny.edu .

     ABSTRACT:   Consider these results:

     (a) [N. Noble,1972] "every G\delta-dense subspace in a product of separable metric spaces is C-embedded";

     (b) [M. Ulmer, 1970/73] "every \Sigma-product in a product of first-countable spaces is C-embedded";

     (c) [R. Pol and E. Pol, 1976, also A. V. Arhangel'ski, 2000, as corollaries of more general theorems] "every dense subset of a product of completely regular, first-countable spaces is C-embedded in its G\delta-closure";

     (d) [M. Ulmer, 1970/73] "every \Sigma-product in a product of P-spaces is C-embedded".

      In this talk, results related to continuous extension of functions defined on subsets of product spaces with the \kappa-box topology will be presented. Here is the case \kappa=\omega of our main theorem, which simultaneously generalizes the above-mentioned results, and a theorem from our joint paper with W. W. Comfort, "Continuous extensions of functions defined on subsets of products" (to appear in Topology Appl., 2012).

      Theorem. Let \alpha be an infinite cardinal and Y be dense in an open subset of XI:=\Pi{i\in I}\,Xi with each Xi a T1-space. Let also q in XI \ Y have the property that for each J\in[I]^{\leq\alpha} there is y in Y such that yJ=qJ and let Z be a regular space with a \overline{G}{\alpha^+}-diagonal. Suppose that for each i in I either \chi(qi,Xi)\leq\alpha or qi is a P-point in Xi.

      Then every continuous f:Y ==> Z extends continuously over Y cup\{q\}.

      Several corollaries and consequences of this theorem will be given.

(Note 1) This is joint work with W. W. Comfort and D. Rivers.

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APRIL 19:   Benjamin Steinberg, City College, CUNY, "Free profinite monoids". At CCNY. Tea 3:15pm, NAC 8/133, talk 4pm, Shepard 20. For more information and parking: R. Kopperman, rdkcc@ccny.cuny.edu

     ABSTRACT:   We give a survey on the motivation for studying free profinite monoids and some recent developments on their structure, including relations to profinite group theory and symbolic dynamics.

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MAY 3:   Steve Matthews, Warwick University, Coventry, UK, "The cost of partial metrics". At CCNY; tea 3:30 p.m., AC8/134, talk 4 p.m., Harris/09. For more information and parking: R. Kopperman, rdkcc@ccny.cuny.edu .

     ABSTRACT:   Partial metric spaces have evolved as an interesting means for approximating metric spaces that are computable. What remains to be done is to enhance this work with a notion of how such computability can be costed. Using Wadge's work on hiatons, we introduce a notion of cost for partial metric spaces.

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MAY 31: Jerzy Kakol, A. Mickiewicz University, Poznan, Poland: "An Amir-Lindenstrauss Type Theorem for Spaces Cp(X) and Applications". At Baruch College, CUNY, Vertical Campus, Lexington Avenue and 24th Street; tea at 3:15 p.m., talk at 4:00 p.m., both in room VC 6-215. For local information, contact A. Todd, aaron.todd@baruch.cuny.edu .

     ABSTRACT:  Using the index of Nagami, we show new topological cardinal inequalities for spaces Cp(\upsilon\lambdaX) to provide a version of an Amir-Lindenstrauss theorem for spaces Cp(X)which in a particular case states that:

     If L embedded in Cp(X) is a Lindelof \Sigma-space and the Nagami index Nag(X) of X less or equal than the density d(L) of L (this holds, for example, if X is a Lindelof \Sigma-space), then

     (i) there exists a completely regular Hausdorff space Y such that Nag(Y) is less than or equal to Nag(X), L embedded in Cp(Y) and d(Y) = d(L);

     (ii) Y admits a weaker completely regular Hausdorff topology \tau' such that \weight(Y,\tau') is less than or equal to d(Y).

      This applies, among other things, to characterize analytic sets for the weak topology of any locally convex space E in a large class G of locally convex spaces that includes, for instance, (DF)-spaces and (LF)-spaces. The latter yields a result of Cascales-Orihuela about weak metrizability of weakly compact sets in spaces in the class G.

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For more information, contact:

CCNY (212-650-5346): R. Kopperman, S. Popvassilev
College of Staten Island (718-982-3626): P. R. Misra
Baruch College (646-312-4136): A. Todd
LIU C. W. Post Center (516-299-2447): S. Andima
Medgar Evers College, (718-270-6416): H. Pajoohesh
Queensborough Community College, (718-281-5291): F. Jordan


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