On the homogeneity of topological spaces.
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Date
2024
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Universidad de Concepción
Abstract
En esta tesis estudiamos la preservación de homogeneidad (y no homogeneidad) de contraejemplos universales no metrizables bajo productos e hiperespacios, con el fin de responder las siguientes preguntas: ¿Es la ω-ésima potencia del plano de Niemytzki homogénea? [Fitzpatrick Jr. and Zhou (1990), Problem 5] y ¿Es el hiperespacio de los cerrados no vacíos de la doble flecha homogéneo? [Arkhangel’skiˇi (1987), Problem II.1].
Para abordar la primera pregunta, investigamos subespacios de la ω-ésima potencia del plano de Niemytzki y la respondemos parcialmente demostrando la homogeneidad del producto entre el plano de Niemytzki y la ω-ésima potencia de un abierto básico. Como consecuencia, concluimos que el producto de la ω-ésima potencia del plano de Niemytzki con la ω-ésima potencia de un abierto básico es también homogéneo.
Para responder a la segunda pregunta, analizamos hiperespacios de la doble flecha y ofrecemos una respuesta parcial probando que los espacios de uniones de a lo más una cantidad finita de intervalos cerrados, así como todos los
productos simétricos excepto el primero, no son homogéneos. Como contraparte, demostramos que el segundo producto simétrico de la recta de Sorgenfrey es homogéneo. Además, logramos dar una imagen completa de cómo lucen los autohomeomorfismos de potencias finitas de la doble flecha. Mostramos que cualquier autohomeomorfismo de una potencia finita de la doble flecha es localmente (fuera de un conjunto nunca denso) un producto de encajes monótonos que van desde un intervalo abierto-cerrado de la doble flecha a esta, seguido de una permutación de las coordenadas.
In this thesis we study the preservation of homogeneity (and non-homogeneity) of nonmetrizable universal counterexamples under products and hyperspaces. The main objective is to answer the following questions: Is ωth power of the Niemytzki plane homogeneous? [Fitzpatrick Jr. and Zhou (1990), Problem 5], Is the hyperspace of nonempty closed subsets of the double arrow homogeneous? [Arkhangel’skiˇi (1987), Problem II.1]. To answer the first question, we study subspaces of the ωth power of the Niemytzki plane and we answer it partially by showing that the product of the ωth power of the Niemytzki plane and the ωth power a basic open is homogeneous. As a consequence, the product of the ωth power of the Niemytzki plane and the ωth power of a basic open is homogeneous. To answer the second question, we study hyperspaces of the double arrow and we answer it partially by showing that the spaces of all unions of at most a finite number of closed intervals and all symmetric products, except the first one, are nonhomogeneous. As a counterpart, we prove that the second symmetric product of the Sorgenfrey line is homogeneous. Moreover, we give a complete picture on how the autohomeomorphisms of finite powers of the double arrow looks like, by showing that any autohomeomorphism of a finite power of the double arrow is locally (outside of a nowhere dense set) a product of monotone embeddings from a clopen interval of the double arrow to the double arrow, followed by a permutation of the coordinates.
In this thesis we study the preservation of homogeneity (and non-homogeneity) of nonmetrizable universal counterexamples under products and hyperspaces. The main objective is to answer the following questions: Is ωth power of the Niemytzki plane homogeneous? [Fitzpatrick Jr. and Zhou (1990), Problem 5], Is the hyperspace of nonempty closed subsets of the double arrow homogeneous? [Arkhangel’skiˇi (1987), Problem II.1]. To answer the first question, we study subspaces of the ωth power of the Niemytzki plane and we answer it partially by showing that the product of the ωth power of the Niemytzki plane and the ωth power a basic open is homogeneous. As a consequence, the product of the ωth power of the Niemytzki plane and the ωth power of a basic open is homogeneous. To answer the second question, we study hyperspaces of the double arrow and we answer it partially by showing that the spaces of all unions of at most a finite number of closed intervals and all symmetric products, except the first one, are nonhomogeneous. As a counterpart, we prove that the second symmetric product of the Sorgenfrey line is homogeneous. Moreover, we give a complete picture on how the autohomeomorphisms of finite powers of the double arrow looks like, by showing that any autohomeomorphism of a finite power of the double arrow is locally (outside of a nowhere dense set) a product of monotone embeddings from a clopen interval of the double arrow to the double arrow, followed by a permutation of the coordinates.
Description
Tesis presentada para optar al grado de Doctor en Matemática.
Keywords
Hiperespacio