Fixed points and coupled fixed points in partially ordered ν-generalized metric spaces

[EN] New fixed point and coupled fixed point theorems in partially ordered ν-generalized metric spaces are presented. Since the product of two ν-generalized metric spaces is not in general a ν-generalized metric space, a different approach is needed than in the case of standard metric spaces.Abtahi, M.; Kadelburg, Z.; Radenovic, S. (2018). Fixed points and coupled fixed points in partially ordered ν-generalized metric spaces. Applied General Topology. 19(2):189-201. doi:10.4995/agt.2018.7409SWORD189201192M. Abtahi, Fixed point theorems for Meir-Keeler type contractions in metric spaces, Fixed Point Theory 17, no. 2 (2016), 225-236.M. Abtahi, Z. Kadelburg and S. Radenovic, Fixed points of Ciric-Matkowski-type contractions in $nu$-generalized metric spaces, Rev. Real Acad. Cienc. Exac. Fis. Nat. Ser. A, Mat. 111, no. 1 (2017), 57-64.B. Alamri, T. Suzuki and L. A. Khan, Caristi's fixed point theorem and Subrahmanyam's fixed point theorem in $nu$-generalized metric spaces, J. Function Spaces, 2015, Art. ID 709391, 6 pp.V. Berinde and M. Pacurar, Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces, Fixed Point Theory Appl. (2012) 2012:115. https://doi.org/10.1186/1687-1812-2012-115T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65, no. 7 (2006), 1379-1393. https://doi.org/10.1016/j.na.2005.10.017A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31-37.Lj. B. Ciric, A new fixed-point theorem for contractive mappings, Publ. Inst. Math. (N.S) 30 (44) (1981), 25-27.Z. Kadelburg and S. Radenovic, On generalized metric spaces: A survey, TWMS J. Pure Appl. Math. 5 (2014), 3-13.Z. Kadelburg and S. Radenovic, Fixed point results in generalized metric spaces without Hausdorff property, Math. Sciences 8:125 (2014). https://doi.org/10.1007/s40096-014-0125-6R. Kannan, Some results on fixed points-II, Amer. Math. Monthly 76 (1969), 405-408.W. A. Kirk and N. Shahzad, Generalized metrics and Caristi's theorem, Fixed Point Theory Appl. 2013:129 (2013). https://doi.org/10.1186/1687-1812-2013-129V. Lakshmikantham and Lj. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009), 4341-4349. https://doi.org/10.1016/j.na.2008.09.020N. V. Luong and N. X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011), 983-992. https://doi.org/10.1016/j.na.2010.09.055J. J. Nieto and R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, Engl. Ser. 23, no. 12 (2007), 2205-2212. https://doi.org/10.1007/s10114-005-0769-0P. D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 64 (2006), 546-557. https://doi.org/10.1016/j.na.2005.04.044B. Samet, Discussion on 'A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces' by A. Branciari, Publ. Math. Debrecen 76 (2010), 493-494.B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517. https://doi.org/10.1016/j.na.2010.02.026I. R. Sarma, J. M. Rao and S. S. Rao, Contractions over generalized metric spaces, J. Nonlinear Sci. Appl. 2 (2009), 180-182. https://doi.org/10.22436/jnsa.002.03.06T. Suzuki, Generalized metric spaces do not have the compatible topology, Abstr. Appl. Anal., 2014, Art. ID 458098, 5 pp.T. Suzuki, B. Alamri and L. A. Khan, Some notes on fixed point theorems in v-generalized metric spaces, Bull. Kyushu Inst. Tech. Pure Appl. Math. 62 (2015), 15-23.M. Turinici, Functional contractions in local Branciari metric spaces, Romai J. 8 (2012),189-199

Lawson topology of the space of formal balls and the hyperbolic topology

AbstractLet (X,d) be a metric space and BX=X×R denote the partially ordered set of (generalized) formal balls in X. We investigate the topological structures of BX, in particular the relations between the Lawson topology and the product topology. We show that the Lawson topology coincides with the product topology if (X,d) is a totally bounded metric space, and show examples of spaces for which the two topologies do not coincide in the spaces of their formal balls. Then, we introduce a hyperbolic topology, which is a topology defined on a metric space other than the metric topology. We show that the hyperbolic topology and the metric topology coincide on X if and only if the Lawson topology and the product topology coincide on BX

Topological Ramsey spaces from Fra\"iss\'e classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fra\"iss\'e classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Soki\v{c} is extended to equivalence relations for finite products of structures from Fra\"iss\'e classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudl\'ak-R\"odl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fra\"iss\'e classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor in \cite{Baumgartner/Taylor78} generating p-points which are $k$-arrow but not $k+1$-arrow, and in a partial order of Blass in \cite{Blass73} producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of $n$ many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra $\mathcal{P}(n)$. If the number of Fra\"iss\'e classes on each block grows without bound, then the Tukey types of the p-points below the space's associated ultrafilter have the structure exactly $[\omega]^{<\omega}$. In contrast, the set of isomorphism types of any product of finitely many Fra\"iss\'e classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.Comment: 35 pages. Abstract and introduction re-written to make very clear the main points of the paper. Some typos and a few minor errors have been fixe

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We show the characterization analogous to dimension groups of partially ordered real vector spaces with interpolation works, but sequential direct limits of simplicial vector spaces only under strong assumptions. We also provide and generalize a proof of a result of Fuchs asserting that the real polynomial algebra with pointwise ordering coming from an interval satisfies Riesz interpolatio

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A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces,the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval $[0,1]$ and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in $[0,1]$

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The canonical map from the Kan subdivision of a product of finite simplicial sets to the product of the Kan subdivisions is a simple map, in the sense that its geometric realization has contractible point inverses