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Jan Van Mill - One of the best experts on this subject based on the ideXlab platform.
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Addendum to "Nearly Countable Dense Homogeneous Spaces"
Canadian Journal of Mathematics, 2014Co-Authors: Michael Hrušáak, Jan Van MillAbstract:This paper provides an addendum to M. Hrusak and J. van Mill “Nearly Countable Dense Homogeneous Spaces.” Canad. J. Math., published online 2013-03-08, http://dx.doi.org/10.4153/ CJM-2013-006-8. It was brought to our attention by Su Gao that the proof of Theorem 5.2 in our paper is incomplete. We are indebted to him for this observation. The aim of this note is to correct this. Theorem 5.2 Let G be a closed subgroup of S∞ and let κ be the number of orbits for the canonical action G × 2N → 2N. Then there is an action of a Polish group H on X = N× [0, 1) such that X has κ H-types of Countable Dense sets. Proof Let G act on X in the following natural way: (g, (n, t)) 7−→ (g(n), t) for g ∈ G, n ∈ N, t ∈ [0, 1). Put F = { f ∈H (X) : (∀ n ∈ N)( f (n, 0) = (n, 0)) } . Then F is a closed normal subgroup of H (X) and hence is Polish. Moreover, for any two Countable Dense Subsets D and E of N × (0, 1) there exists f ∈ F such that f (D) = E. Treat G also as subgroup of H (X). The Polish semi-direct product group H = G o F acts on X as follows: ((g, f ), x) 7→ ( f ◦ g)(x) for f ∈ F, g ∈ G, x ∈ X. Note that topologically, H = G o F is G× F, but its group operation ∗ is given by (g1, f1) ∗ (g2, f2) = (g1g2, f1g1 f2g 1 ). A typical Countable Dense Subset of X has the form D ∪ A, where D is a Countable Dense Subset of N× (0, 1), and A ⊆ N× {0}. By identifying P(N× {0}) and 2N in the standard way, it is clear that we get what we want. Centro de Ciencias Matematicas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacan, 58089, Mexico e-mail: michael@matmor.unam.mx Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands e-mail: j.van.mill@vu.nl Received by the editors October 28, 2013. Published electronically November 7, 2013. The first author was supported by a PAPIIT grant IN 102311 and CONACyT grant 177758. The second author is pleased to thank the Centro de Ciencias Matematicas at Morelia for generous hospitality and support. AMS subject classification: 54H05, 03E15, 54E50.
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Erdos Space and Homeomorphism Groups of Manifolds
2010Co-Authors: Jan J. Dijkstra, Jan Van MillAbstract:Let M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary Countable Dense Subset of M. Consider the topological group \mathcal{H}(M,D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. The authors present a complete solution to the topological classification problem for \mathcal{H}(M,D) as follows. If M is a one-dimensional topological manifold, then they proved in an earlier paper that \mathcal{H}(M,D) is homeomorphic to \mathbb{Q}^\omega, the Countable power of the space of rational numbers. In all other cases they find in this paper that \mathcal{H}(M,D) is homeomorphic to the famed Erdos space \mathfrak E, which consists of the vectors in Hilbert space \ell^2 with rational coordinates. They obtain the second result by developing topological characterizations of Erdos space.
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Homeomorphism groups of manifolds and Erdos space
Electronic Research Announcements of the American Mathematical Society, 2004Co-Authors: Jan J. Dijkstra, Jan Van MillAbstract:Let M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary Countable Dense Subset of M. Consider the topological group H(M, D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H(M, D) as follows. If M is a one-dimensional topological manifold, then H(M, D) is homeomorphic to ℚ
Jerzy Zabczyk - One of the best experts on this subject based on the ideXlab platform.
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COMPLETENESS OF BOND MARKET DRIVEN BY LÉVY PROCESS
International Journal of Theoretical and Applied Finance, 2010Co-Authors: Michał Barski, Jerzy ZabczykAbstract:The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a Countable, Dense Subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the Levy measure consists of a finite number of points. Explicit constructions of contingent claims which cannot be replicated are provided.
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Completeness of bond market driven by L\'evy process
arXiv: Probability, 2008Co-Authors: Michał Barski, Jerzy ZabczykAbstract:The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a Countable, Dense Subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the L\'evy measure consists of a finite number of points. Explicit constructions of contingent claims which can not be replicated are provided.
Michał Barski - One of the best experts on this subject based on the ideXlab platform.
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COMPLETENESS OF BOND MARKET DRIVEN BY LÉVY PROCESS
International Journal of Theoretical and Applied Finance, 2010Co-Authors: Michał Barski, Jerzy ZabczykAbstract:The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a Countable, Dense Subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the Levy measure consists of a finite number of points. Explicit constructions of contingent claims which cannot be replicated are provided.
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Completeness of bond market driven by L\'evy process
arXiv: Probability, 2008Co-Authors: Michał Barski, Jerzy ZabczykAbstract:The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a Countable, Dense Subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the L\'evy measure consists of a finite number of points. Explicit constructions of contingent claims which can not be replicated are provided.
Michael Hrušák - One of the best experts on this subject based on the ideXlab platform.
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A disjointly tight irresolvable space
Applied General Topology, 2020Co-Authors: Angelo Bella, Michael HrušákAbstract:In this short note we prove the existence (in ZFC) of a completely regular Countable disjointly tight irresolvable space by showing that every sub-maximal Countable Dense Subset of 2c is disjointly tight.
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CARDINAL INVARIANTS RELATED TO SEQUENTIAL SEPARABILITY
2008Co-Authors: Michael Hrušák, Juris SteprānsAbstract:Cardinal invariants related to sequential separability of generalized Cantor cubes 2κ, introduced by M. Matveev, are studied here. In particular, it is shown that the following assertions are relatively consistent with ZFC: (1) 2ω1 is sequentially separable, yet there is a Countable Dense Subset of 2ω1 containing no non-trivial convergent subsequence, (2) 2ω1 is not sequentially separable, yet it is sequentially compact. The work contained in this paper is devoted to studying combinatorial properties of independent families and their relationship with sequential separability of generalized Cantor cubes. Connections with Q-sets and hence the existence of separable non-metrizable Moore spaces is also mentioned. A topological space X is sequentially separable if there is a Countable D ⊆ X such that for every x ∈ X there is a sequence {xn : n ∈ ω} ⊆ D converging to x; such a D ⊆ X will be called sequentially Dense in X. A space X is strongly sequentially separable if it is separable and every Countable Dense Subset of X is sequentially Dense. Here we consider sequential separability of 2 equipped with the product topology. Recall that a set A is a pseudo-intersection of a family F ⊆ [ω] if A ⊆∗ F for every F ∈ F and, F is centered if every non-empty finite subfamily of F has an infinite intersection. A family S ⊆ [ω] is splitting if ∀A ∈ [ω] ∃S ∈ S |A ∩ S| = |A \ S| = ω. A family I ⊆ [ω] is independent provided that for every nonempty disjoint F1,F2 ∈ [I]
Michael Hrušáak - One of the best experts on this subject based on the ideXlab platform.
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Addendum to "Nearly Countable Dense Homogeneous Spaces"
Canadian Journal of Mathematics, 2014Co-Authors: Michael Hrušáak, Jan Van MillAbstract:This paper provides an addendum to M. Hrusak and J. van Mill “Nearly Countable Dense Homogeneous Spaces.” Canad. J. Math., published online 2013-03-08, http://dx.doi.org/10.4153/ CJM-2013-006-8. It was brought to our attention by Su Gao that the proof of Theorem 5.2 in our paper is incomplete. We are indebted to him for this observation. The aim of this note is to correct this. Theorem 5.2 Let G be a closed subgroup of S∞ and let κ be the number of orbits for the canonical action G × 2N → 2N. Then there is an action of a Polish group H on X = N× [0, 1) such that X has κ H-types of Countable Dense sets. Proof Let G act on X in the following natural way: (g, (n, t)) 7−→ (g(n), t) for g ∈ G, n ∈ N, t ∈ [0, 1). Put F = { f ∈H (X) : (∀ n ∈ N)( f (n, 0) = (n, 0)) } . Then F is a closed normal subgroup of H (X) and hence is Polish. Moreover, for any two Countable Dense Subsets D and E of N × (0, 1) there exists f ∈ F such that f (D) = E. Treat G also as subgroup of H (X). The Polish semi-direct product group H = G o F acts on X as follows: ((g, f ), x) 7→ ( f ◦ g)(x) for f ∈ F, g ∈ G, x ∈ X. Note that topologically, H = G o F is G× F, but its group operation ∗ is given by (g1, f1) ∗ (g2, f2) = (g1g2, f1g1 f2g 1 ). A typical Countable Dense Subset of X has the form D ∪ A, where D is a Countable Dense Subset of N× (0, 1), and A ⊆ N× {0}. By identifying P(N× {0}) and 2N in the standard way, it is clear that we get what we want. Centro de Ciencias Matematicas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacan, 58089, Mexico e-mail: michael@matmor.unam.mx Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands e-mail: j.van.mill@vu.nl Received by the editors October 28, 2013. Published electronically November 7, 2013. The first author was supported by a PAPIIT grant IN 102311 and CONACyT grant 177758. The second author is pleased to thank the Centro de Ciencias Matematicas at Morelia for generous hospitality and support. AMS subject classification: 54H05, 03E15, 54E50.
