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Papers by carlos uzcategui

Research paper thumbnail of Polish globalization of Polish group partial actions

Mathematical Logic Quarterly, 2017

Let X be a separable metrizable space. We establish a criteria for the existence of a metrizable ... more Let X be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group G on X. If G and X are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups.

Research paper thumbnail of Borel globalizations of partial actions of Polish groups

Archive for Mathematical Logic, 2017

We show that the enveloping space XG of a partial action of a Polish group G on a Polish space X ... more We show that the enveloping space XG of a partial action of a Polish group G on a Polish space X is a standard Borel space, that is to say, there is a topology τ on XG such that (XG, τ) is Polish and the quotient Borel structure on XG is equal to Borel(XG, τ). To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups.

Research paper thumbnail of On the Ellis semigroup of a distal dynamical systems on a compact metric countable space

Let X be a compact metric countable space, f : X → X be a homeomorphism and E(X, f) its Ellis sem... more Let X be a compact metric countable space, f : X → X be a homeomorphism and E(X, f) its Ellis semigroup. Among other results we show that the following statements are equivalent: (i) (X, f) is equicontinuous, (ii) (X, f) is distal and (iii) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group. We also show, for spaces with finitely many limit points, that E(X, f) is abelian if, and only if, every function in E(X, f) is continuous (i.e. the system is WAP).

Research paper thumbnail of On the Ellis semigroup of a cascade on a compact metric countable space

Semigroup Forum, 2020

Let X be a compact metric countable space, let f : X → X be a homeomorphism and let E(X, f) be it... more Let X be a compact metric countable space, let f : X → X be a homeomorphism and let E(X, f) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (i) (X, f) is equicontinuous, (ii) (X, f) is distal and (iii) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group.

Research paper thumbnail of Ideal games and Ramsey sets

Proceedings of the American Mathematical Society, 2012

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal... more It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a coideal H is semiselective if and only if Matet's game-theoretic characterization of the H-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory, and gives a game-theoretic counterpart to a theorem of Farah, asserting that a co-ideal H is semiselective if and only if the family of H-Ramsey subsets of N [∞] coincides with the family of those sets having the abstract H-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal H all sets of real numbers are H-Ramsey.

Research paper thumbnail of Normal filters generated by a family of sets

Proceedings of the American Mathematical Society, 1987

We study normal filters on the set spaces λ , P κ ( λ ) , [ λ ] κ \lambda ,{\mathcal {P}_\kappa }... more We study normal filters on the set spaces λ , P κ ( λ ) , [ λ ] κ \lambda ,{\mathcal {P}_\kappa }\left ( \lambda \right ),{\left [ \lambda \right ]^\kappa } , and ( λ ) κ {\left ( \lambda \right )^\kappa } . We characterize the least normal γ \gamma -complete filter containing a given γ \gamma -complete filter for γ ≥ ω 1 \gamma \geq {\omega _1} . If F \mathcal {F} is a ω 1 {\omega _1} -complete filter on any of the set spaces mentioned, the least ω 1 {\omega _1} -complete normal filter containing it is the filter generated by the sets { x ∈ E | α 1 , … , α n ∈ x → x ∈ f ( α 1 , … , α n ) } \left \{ {x \in E\left | {{\alpha _1}, \ldots ,{\alpha _n} \in x \to x \in f\left ( {{\alpha _1}, \ldots ,{\alpha _n}} \right )} \right .} \right \} where f : λ > ω → F f:{\lambda ^{ > \omega }} \to \mathcal {F} .

Research paper thumbnail of Bases and selectors for cofinal families of countable sets

Non UBCUnreviewedAuthor affiliation: Universidad Industrial de SantanderFacult

Research paper thumbnail of Polish globalization of Polish group partial actions

Mathematical Logic Quarterly, 2017

Let X be a separable metrizable space. We establish a criteria for the existence of a metrizable ... more Let X be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group G on X. If G and X are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups.

Research paper thumbnail of Borel globalizations of partial actions of Polish groups

Archive for Mathematical Logic, 2017

We show that the enveloping space XG of a partial action of a Polish group G on a Polish space X ... more We show that the enveloping space XG of a partial action of a Polish group G on a Polish space X is a standard Borel space, that is to say, there is a topology τ on XG such that (XG, τ) is Polish and the quotient Borel structure on XG is equal to Borel(XG, τ). To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups.

Research paper thumbnail of On the Ellis semigroup of a distal dynamical systems on a compact metric countable space

Let X be a compact metric countable space, f : X → X be a homeomorphism and E(X, f) its Ellis sem... more Let X be a compact metric countable space, f : X → X be a homeomorphism and E(X, f) its Ellis semigroup. Among other results we show that the following statements are equivalent: (i) (X, f) is equicontinuous, (ii) (X, f) is distal and (iii) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group. We also show, for spaces with finitely many limit points, that E(X, f) is abelian if, and only if, every function in E(X, f) is continuous (i.e. the system is WAP).

Research paper thumbnail of On the Ellis semigroup of a cascade on a compact metric countable space

Semigroup Forum, 2020

Let X be a compact metric countable space, let f : X → X be a homeomorphism and let E(X, f) be it... more Let X be a compact metric countable space, let f : X → X be a homeomorphism and let E(X, f) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (i) (X, f) is equicontinuous, (ii) (X, f) is distal and (iii) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group.

Research paper thumbnail of Ideal games and Ramsey sets

Proceedings of the American Mathematical Society, 2012

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal... more It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a coideal H is semiselective if and only if Matet's game-theoretic characterization of the H-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory, and gives a game-theoretic counterpart to a theorem of Farah, asserting that a co-ideal H is semiselective if and only if the family of H-Ramsey subsets of N [∞] coincides with the family of those sets having the abstract H-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal H all sets of real numbers are H-Ramsey.

Research paper thumbnail of Normal filters generated by a family of sets

Proceedings of the American Mathematical Society, 1987

We study normal filters on the set spaces λ , P κ ( λ ) , [ λ ] κ \lambda ,{\mathcal {P}_\kappa }... more We study normal filters on the set spaces λ , P κ ( λ ) , [ λ ] κ \lambda ,{\mathcal {P}_\kappa }\left ( \lambda \right ),{\left [ \lambda \right ]^\kappa } , and ( λ ) κ {\left ( \lambda \right )^\kappa } . We characterize the least normal γ \gamma -complete filter containing a given γ \gamma -complete filter for γ ≥ ω 1 \gamma \geq {\omega _1} . If F \mathcal {F} is a ω 1 {\omega _1} -complete filter on any of the set spaces mentioned, the least ω 1 {\omega _1} -complete normal filter containing it is the filter generated by the sets { x ∈ E | α 1 , … , α n ∈ x → x ∈ f ( α 1 , … , α n ) } \left \{ {x \in E\left | {{\alpha _1}, \ldots ,{\alpha _n} \in x \to x \in f\left ( {{\alpha _1}, \ldots ,{\alpha _n}} \right )} \right .} \right \} where f : λ > ω → F f:{\lambda ^{ > \omega }} \to \mathcal {F} .

Research paper thumbnail of Bases and selectors for cofinal families of countable sets

Non UBCUnreviewedAuthor affiliation: Universidad Industrial de SantanderFacult