Squares of function spaces and function spaces on squares Miko laj - PowerPoint PPT Presentation

Squares of function spaces and function spaces on squares Miko� laj Krupski University of Warsaw TOPOSYM, 2016 Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

For a Tychonoff space X , C p ( X ) is the space of continuous real-valued functions on X , with the pointwise topology. Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

For a Tychonoff space X , C p ( X ) is the space of continuous real-valued functions on X , with the pointwise topology. Borsuk-Dugundji Extension Theorem If X is metrizable and A ⊆ X is closed, then there exists a linear continuous function φ : C p ( A ) → C p ( X ) such that φ ( f ) ↾ A = f , for any f ∈ C p ( A ). Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

For a Tychonoff space X , C p ( X ) is the space of continuous real-valued functions on X , with the pointwise topology. Borsuk-Dugundji Extension Theorem If X is metrizable and A ⊆ X is closed, then there exists a linear continuous function φ : C p ( A ) → C p ( X ) such that φ ( f ) ↾ A = f , for any f ∈ C p ( A ). Corollary If X is metrizable and A ⊆ X is closed, then C p ( X ) ≈ C p ( A ) × { f ∈ C p ( X ): f ↾ A = 0 } ≈ C p ( A ) × C p ( X / A ) Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

For a Tychonoff space X , C p ( X ) is the space of continuous real-valued functions on X , with the pointwise topology. Borsuk-Dugundji Extension Theorem If X is metrizable and A ⊆ X is closed, then there exists a linear continuous function φ : C p ( A ) → C p ( X ) such that φ ( f ) ↾ A = f , for any f ∈ C p ( A ). Corollary If X is metrizable and A ⊆ X is closed, then C p ( X ) ≈ C p ( A ) × { f ∈ C p ( X ): f ↾ A = 0 } ≈ C p ( A ) × C p ( X / A ) It follows that, e.g. C p ([0 , 1]) ≈ C p ([0 , 1]) × C p ([0 , 1]) C p ( R ) ≈ C p ( R ) × C p ( R ) Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii), 1978, 1990 Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is an infinite ’nice’ topological space, e.g. is compact or metrizable? Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii), 1978, 1990 Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is an infinite ’nice’ topological space, e.g. is compact or metrizable? Some motivations: Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii), 1978, 1990 Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is an infinite ’nice’ topological space, e.g. is compact or metrizable? Some motivations: Related to a more general question: Does C p ( X ) space has ’good’ factorization properties? Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii), 1978, 1990 Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is an infinite ’nice’ topological space, e.g. is compact or metrizable? Some motivations: Related to a more general question: Does C p ( X ) space has ’good’ factorization properties? Many natural examples of infinite-dimensional linear topological spaces possess good factorization properties. Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii), 1978, 1990 Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is an infinite ’nice’ topological space, e.g. is compact or metrizable? Some motivations: Related to a more general question: Does C p ( X ) space has ’good’ factorization properties? Many natural examples of infinite-dimensional linear topological spaces possess good factorization properties. Factorization properties help constructing homeomorphisms between function spaces. Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii), 1978, 1990 Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is an infinite ’nice’ topological space, e.g. is compact or metrizable? Some motivations: Related to a more general question: Does C p ( X ) space has ’good’ factorization properties? Many natural examples of infinite-dimensional linear topological spaces possess good factorization properties. Factorization properties help constructing homeomorphisms between function spaces. Related to another important question: Which topological properties of C p ( X ) are productive? Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii), 1978, 1990 Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is an infinite ’nice’ topological space, e.g. is compact or metrizable? Some motivations: Related to a more general question: Does C p ( X ) space has ’good’ factorization properties? Many natural examples of infinite-dimensional linear topological spaces possess good factorization properties. Factorization properties help constructing homeomorphisms between function spaces. Related to another important question: Which topological properties of C p ( X ) are productive? Open question: Suppose that C p ( X ) is Lindel¨ of. Is it true that C p ( X ) × C p ( X ) is Lindel¨ of? Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1978) Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is infinite compact? Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1978) Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is infinite compact? Theorem (Gul’ko / Marciszewski, 1988) No, there exists an infinite compact (nonmetrizable) space X such that C p ( X ) is not homeomorphic to C p ( X ) × C p ( X ). Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1978) Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is infinite compact? Theorem (Gul’ko / Marciszewski, 1988) No, there exists an infinite compact (nonmetrizable) space X such that C p ( X ) is not homeomorphic to C p ( X ) × C p ( X ). Gul’ko example Consider X = [0 , ω 1 ], then C p ( X ) �≈ C p ( X ) × C p ( X ). Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1978) Is it true that C p ( X ) is homeomorphic to C p ( X ) × C p ( X ) provided X is infinite compact? Theorem (Gul’ko / Marciszewski, 1988) No, there exists an infinite compact (nonmetrizable) space X such that C p ( X ) is not homeomorphic to C p ( X ) × C p ( X ). Gul’ko example Consider X = [0 , ω 1 ], then C p ( X ) �≈ C p ( X ) × C p ( X ). Marciszewski example X = ω ∪ { p A : A ∈ A} ∪ {∞} , where A is a suitable almost disjoint family on ω . Points in ω are isolated, neighborhoods of p A are of the form { p A } ∪ ( A \ F ), where F is finite. Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1990) Is it true C p ( X ) is (linearly) homeomorphic to C p ( X ) × C p ( X ) provided X is infinite metrizable? Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1990) Is it true C p ( X ) is (linearly) homeomorphic to C p ( X ) × C p ( X ) provided X is infinite metrizable? Theorem (Pol, 1995) There is an infinite metrizable (compact) space X with C p ( X ) not linearly homeomorphic to C p ( X ) × C p ( X ). Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1990) Is it true C p ( X ) is (linearly) homeomorphic to C p ( X ) × C p ( X ) provided X is infinite metrizable? Theorem (Pol, 1995) There is an infinite metrizable (compact) space X with C p ( X ) not linearly homeomorphic to C p ( X ) × C p ( X ). Theorem (van Mill, Pelant, Pol, 2003) There is an infinite metrizable (compact) space X with C p ( X ) not uniformly homeomorphic to C p ( X ) × C p ( X ). Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares

Problem (Arhangel’skii, 1990) Is it true C p ( X ) is (linearly) homeomorphic to C p ( X ) × C p ( X ) provided X is infinite metrizable? Theorem (Pol, 1995) There is an infinite metrizable (compact) space X with C p ( X ) not linearly homeomorphic to C p ( X ) × C p ( X ). Theorem (van Mill, Pelant, Pol, 2003) There is an infinite metrizable (compact) space X with C p ( X ) not uniformly homeomorphic to C p ( X ) × C p ( X ). van Mill, Pelant, Pol example X = Cook continuum Miko� laj Krupski University of Warsaw Squares of function spaces and function spaces on squares