Topological aspects of enumeration degrees Takayuki Kihara Department of Mathematics, University of California, Berkeley, USA Joint Work with Steffen Lempp, Keng Meng Ng, and Arno Pauly Dagstuhl Seminar on Computability Theory, Feb 20, 2017. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Observation The enumeration degrees = The degrees of points in second-countable T 0 spaces. e -degrees cototal degrees continuous degrees total degrees T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Observation The enumeration degrees = The degrees of points in second-countable T 0 spaces. e -degrees cototal degrees continuous [0 ; 1] ! -degrees degrees total degrees 2 ! -degrees Total degrees = degrees of points in 2 ω . Continuous degrees = degrees of points in [ 0 , 1 ] ω . T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Observation The enumeration degrees = The degrees of points in second-countable T 0 spaces. e -degrees cototal ( X , β )-degrees degrees continuous [0 ; 1] ! -degrees degrees total degrees 2 ! -degrees To each T 0 space X with an enumeration β of a countable basis, one can assign a substructure D ( X , β ) of the e -degrees. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Observation The enumeration degrees = The degrees of points in second-countable T 0 spaces. e -degrees cototal degrees continuous metrizable-degrees metrizable degrees [ fi nite-dimensional [ fi nite-dimensional total degrees metrizable]-degrees metrizable] Total degrees = degrees of points in finite-dimensional metrizable spaces. Continuous degrees = degrees of points in metrizable spaces. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Observation The enumeration degrees = The degrees of points in second-countable T 0 spaces. e -degrees Frechet-degrees ( T 1 ) Hausdor ff -degrees ( T 2 ) Urysohn-degrees ( T 2 : 5 ) continuous metrizable-degrees degrees [ fi nite-dimensional total degrees metrizable]-degrees The e -degrees can be classified in terms of general topology! T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Observation The enumeration degrees = The degrees of points in second-countable T 0 spaces. The e -degrees can be classified in terms of general topology! Total degrees = finite dimensional metrizable e -degrees. Continuous degrees = metrizable e -degrees. (with Madison) Cototal degrees = e -degrees in G δ -spaces. Graph-cototal degrees = e -degrees in ( ω cof ) ω , where ω cof is the set ω equipped with the cofinite topology . Semi-recursive degrees = e -degrees in R with the lower topology. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
To each T 0 space X with a countable basis β , one can assign a substructure D ( X , β ) of the e -degrees. Example (Hausdorff e -degrees) An e -degree d is double-origin if d contains a set of the form: ( X ⊕ X ) ⊕ ( A ∪ P ) ⊕ ( B ∪ N ) , where P and N are X -c.e., A ∪ B is X -co-c.e., and A , B , P , and N are pairwise disjoint. Remark: every 3 -c.e. e -degree is double-origin. Let X be the rational disk endowed with the double origin topology. The degree structure of X ω = the double-origin e -degrees. Since X ω is Hausdorff, all double-origin e -degrees are Hausdorff. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Project 1 Determine the degree structures of second-countable T 0 -spaces which appear in the book “ Counterexamples in Topology [1] ( CiT ).” For most second-countable T 0 spaces X ∈ CiT , + X is very very effective. − The degree structure of X itself is not so interesting. + However, that of its countable product X ω is interesting! [1] L. A. Steen and J. A. Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Current Status of Project 1 total [ ∗ ; Π 0 1 ; Π 0 co-d-CEA 1 ]-SepA double-origin ∆ 0 2 -EvenA ∆ 0 2 -above cylinder-cototal [ ∗ ; ∗ ; Π 0 [ ∆ 0 2 ; ∆ 0 2 ; ∆ 0 Σ 0 telograph-cototal 1 ]-SepA 2 ]-SepA 2 -above graph-cototal ∆ 0 cototal 2 -DBiA T 2 . 5 : irregular lattice space (co-d-CEA), Arens square ( ∆ 0 2 -DBiA), Roy’s lattice space ( ∆ 0 2 -EvenA). T 2 : double origin topology (double-origin). T 1 : cofinite topology (graph-cototal), cocylinder topology (cylinder-cototal), telophase topology (telograph-cototal). T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
To each T 0 space X with a countable basis β = ( B e ) e ∈ ω , one can assign a substructure D ( X , β ) of the e -degrees. Definition The degree of x ∈ X is defined by the e -degree of its coded neighborhood filter: Nbase β ( x ) = { e ∈ ω : x ∈ B e } . Then, the degree structure of X (relative to β ) is defined by D ( X , β ) = { deg e ( Nbase β ( x )) : x ∈ X} . T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
To each T 0 space X with a countable basis β = ( B e ) e ∈ ω , one can assign a substructure D ( X , β ) of the e -degrees. Definition The degree of x ∈ X is defined by the e -degree of its coded neighborhood filter: Nbase β ( x ) = { e ∈ ω : x ∈ B e } . Then, the degree structure of X (relative to β ) is defined by D ( X , β ) = { deg e ( Nbase β ( x )) : x ∈ X} . One can assign degree structures to certain non-second-countable spaces (only using computability on ω , without using α -recursion, E -recursion, ITTM, etc) [E.g. Arhangel’skii (1959) introduced the notion of a network in general topology. Use a countable cs-network to define the degree structure as in Schr¨ oder (2002)] But, if a space is second-countable, then it coincides with the above definition. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Nbase β ( x ) = { e ∈ ω : x ∈ B e } . D ( X , β ) = { deg e ( Nbase β ( x )) : x ∈ X} . Example (Hausdorff e -degrees) The relatively prime integer topology on the positive integers Z > 0 is generated by U b ( a ) = { a + bt : t ∈ Z } , where a and b are relatively prime. Then, for x ∈ Z ω > 0 , Nbase ( x ) = {⟨ n , a , b ⟩ : ( ∃ t ∈ Z ) x ( n ) = a + bt } . T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Nbase β ( x ) = { e ∈ ω : x ∈ B e } . D ( X , β ) = { deg e ( Nbase β ( x )) : x ∈ X} . Basic Idea (De Brecht-K.-Pauly; at Dagstuhl) P : a topological property (e.g. metrizable, Hausdorff, regular) An e -degree d is P if d ∈ D ( X , β ) for some “effective P ” 1 space ( X , β ) . An e -degree d is P -quasiminimal if for any effective P space 2 ( X , β ) , ( ∀ a ) [ a ≤ d & a ∈ D ( X , β ) = ⇒ a = 0 ]. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
T 3 : Cantor space, Euclidean space, Hilbert cube. T 2 . 5 : irregular lattice space, Arens square, Roy’s lattice space, Gandy-Harrington topology. T 2 : double origin topology, relatively prime integer topology. T 1 : cofinite topology, cocylinder topology, telophase topology. T 0 : lower topology, Sierpi´ nski space. Project 2 Given m < n , construct a T n -degree which is T m -quasiminimal! T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
T 3 -degrees vs. T 2 . 5 -degrees. A space is T 3 if it is regular Hausdorff, that is, given any point and closed set are separated by nbhds. A space is T 2 . 5 if any two distinct points are separated by closed nbhds. T 3 : Cantor space, Euclidean space, Hilbert cube. T 2 . 5 : irregular lattice space, Arens square, Roy’s lattice space, Gandy-Harrington topology. Let L be the irregular lattice space. D ( L ω ) = “ 3 -c.e. above total degrees” (Folklore) There is a quasiminimal 3 -c.e. e -degree. (Corollary) There is a T 2 . 5 -degree which is ( T 3 -)quasiminimal. T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
T 2 . 5 -degrees vs. T 2 -degrees. A space is T 2 . 5 if any two distinct points are separated by closed nbhds. A space is T 2 if any two distinct points are separated by open nbhds. T 2 . 5 : irregular lattice space, Arens square, Roy’s lattice space, Gandy-Harrington topology. T 2 : double origin topology, relatively prime integer topology. Theorem Let P be the set Z > 0 endowed with the relatively prime integer topology. ( X n , β n ) n ∈ ω : a countable collection of T 2 . 5 -spaces. D ( P ω ) ⊈ ∪ n ∈ ω D ( X n , β n ) . 1 A sufficiently generic point in P ω is ( T 3 -)quasiminimal. 2 (Open Question): Does there exist a T 2 . 5 -quasiminimal T 2 -degree? T. Kihara, S. Lempp, K. M. Ng, and A. Pauly Topological aspects of enumeration degrees
Recommend
More recommend
