Introduction Heredity of AD-classes References Hereditary, additive and divisible classes in epireflective subcategories of Top Martin Sleziak 10. augusta 2006 Martin Sleziak HAD-classes in epireflective subcategories of Top
Basic definitions Introduction Hereditary coreflective subcategories of Top Heredity of AD-classes A generalization – epireflective subcategories References AD-classes and HAD-classes Subcategories of Top All subcategories are assumed to be full and isomorphism-closed. subcategory of Top = class of topological spaces closer under homeomorphisms subcategory of Top class of spaces closed under coreflective quotients and topological sums epireflective subspaces and products hereditary subspaces Martin Sleziak HAD-classes in epireflective subcategories of Top
Basic definitions Introduction Hereditary coreflective subcategories of Top Heredity of AD-classes A generalization – epireflective subcategories References AD-classes and HAD-classes Hereditary coreflective subcategories of Top The study of such subcategories was suggested by H. Herrlich and M. Hušek in [HH]. Theorem ([Č1, Theorem 3.7]) Let C be a coreflective subcategory of Top with FG ⊆ C . The subcategory C is hereditary if and only if for each X ∈ C all prime factors of X belong to C ( i.e., C is closed under the formation of prime factors ) . prime factor X a = the space with the same neighborhoods of a as X an with all other points isolated Martin Sleziak HAD-classes in epireflective subcategories of Top
Basic definitions Introduction Hereditary coreflective subcategories of Top Heredity of AD-classes A generalization – epireflective subcategories References AD-classes and HAD-classes X a a X α α B ( α ) C ( α ) · · · · · · Figure illustrating the construction of prime factor The construction used in the proof of Theorem 1 Martin Sleziak HAD-classes in epireflective subcategories of Top
Basic definitions Introduction Hereditary coreflective subcategories of Top Heredity of AD-classes A generalization – epireflective subcategories References AD-classes and HAD-classes The same problem can be studied in an epireflective subcategory A of Top . Theorem ([Č2, Theorem 1]) If A is an epireflective subcategory of Top with I 2 / ∈ A and C is a coreflective subcategory of A , then C is hereditary if and only if C is closed under the formation of prime factors. coreflective subcategory in A = a subclass of A closed under the topological sums and the A -extremal epimorphisms quotient map in Top ⇒ A -extremal epimorphism A -extremal epimorphism �⇒ quotient map in Top Martin Sleziak HAD-classes in epireflective subcategories of Top
Basic definitions Introduction Hereditary coreflective subcategories of Top Heredity of AD-classes A generalization – epireflective subcategories References AD-classes and HAD-classes AD-classes and HAD-classes Let A be an epireflective subcategory in Top . We say that a class C ⊆ A is ◮ divisible in A if for each C ∈ C and a quotient map q : C → D with D ∈ A we have D ∈ C . ◮ additive if it is closed under topological sums. AD-class in A = additive and divisible in A HAD-class in A = hereditary AD-class in A Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories Main question Easy to show: AD-class is closed under prime factors ⇒ it is hereditary Question: Is every HAD-class closed under prime factors? Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories The operation △ Definition If X and Y are topological spaces, b ∈ Y and { b } is closed but not open in Y , then we denote by X △ b Y the topological space on the set X × Y which has the final topology w.r.t the family of maps { f , g a ; a ∈ X } , where f : X → X × Y , f ( x ) = ( x , b ) and g a : Y → X × Y , g a ( y ) = ( a , y ) . In other words, we attached the space Y to each point of X by the point b . Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories X a ( a, b ) b V x U Y y Figure: The space X △ b Y Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories Let X a ( Y , b ) = the subspace of X △ b Y on the subset { ( a , b ) } ∪ ( X \ { a } ) × ( Y \ { b } ) Y = any space in which the subset { b } is closed but not open q : X a ( Y , b ) → X a given by q ( x , y ) = x is a quotient map Proposition Let B be an HAD-class in an epireflective subcategory A of Top with I 2 / ∈ A . Let for any X ∈ B there exist Y ∈ B and a non-isolated point b ∈ Y with { b } being closed but not open in Y such that X △ b Y belongs to A . Then B is closed under the formation of prime factors. Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories X a X a a b b Y Y X a ( Y,b ) The subspace X a The quotient map X a ( Y , b ) → X a ( Y , b ) Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories Sufficient conditions Let A be an epireflective subcategory of Top with I 2 / ∈ A and B be an HAD-class in A . If some of the following conditions is fulfilled ◮ A is closed under the operation △ ◮ B contains a prime space, ◮ B contains a nondiscrete zero-dimensional space, ◮ B contains an infinite space with cofinite topology, ◮ A ⊆ Haus , then B is closed under the formation of prime factors. Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories Applications Lemma Let A be an epireflective subcategory of Top such that I 2 / ∈ A . If B = HAD A ( D ) , where D ⊆ A is a set of spaces and B contains at least one prime space, then there exists a prime space B ∈ A such that B = HAD A ( B ) = AD A ( B ) . Moreover, CH ( B ) = HCH ( B ) is hereditary. HAD A ( D ) = HAD-hull of D = the smallest HAD-class in A containing D Theorem Let A be an epireflective subcategory of Top such that I 2 / ∈ A . If B is an HAD-class in A and B contains at least one prime space, then the coreflective hull CH ( B ) of B in Top is hereditary. Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories Bireflective subcategories of Top epireflective subcategories of Top with I 2 / ∈ A = the epireflective subcategories of Top which are closed under the formation of prime factors = epireflective subcategories with A ⊆ Top 0 epireflective subcategories of Top with I 2 ∈ A = bireflective subcategories of Top One-to-one correspondence ([M]): A �→ A ∩ Top 0 Martin Sleziak HAD-classes in epireflective subcategories of Top
Heredity and prime factors Introduction The operation △ Heredity of AD-classes Sufficient conditions References Applications Bireflective subcategories Thanks for your attention! The preprint [S] presented here, as well as the text of this talk and these slides can be found at: http://thales.doa.fmph.uniba.sk/sleziak/papers/ Email: sleziak@fmph.uniba.sk Martin Sleziak HAD-classes in epireflective subcategories of Top
Introduction Heredity of AD-classes References J. Činčura. Heredity and coreflective subcategories of the category of topological spaces. Appl. Categ. Structures , 9:131–138, 2001. J. Činčura. Hereditary coreflective subcategories of categories of topological spaces. Appl. Categ. Structures , 13:329–342, 2005. H. Herrlich and M. Hušek. Some open categorical problems in Top. Appl. Categ. Structures , 1:1–19, 1993. T. Marny. On epireflective subcategories of topological categories. General Topology Appl. , 10:175–181, 1979. Martin Sleziak HAD-classes in epireflective subcategories of Top
Introduction Heredity of AD-classes References M. Sleziak. Hereditary, additive and divisible classes in epireflective subcategories of Top. submitted, 2006. Martin Sleziak HAD-classes in epireflective subcategories of Top
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