Hereditary, additive and divisible classes in epireflective - - PowerPoint PPT Presentation

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

Introduction Heredity of AD-classes References Basic definitions Hereditary coreflective subcategories of Top A generalization – epireflective subcategories 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

Introduction Heredity of AD-classes References Basic definitions Hereditary coreflective subcategories of Top A generalization – epireflective subcategories 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 Xa = 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

Introduction Heredity of AD-classes References Basic definitions Hereditary coreflective subcategories of Top A generalization – epireflective subcategories AD-classes and HAD-classes

· · · α B(α) · · · 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

Introduction Heredity of AD-classes References Basic definitions Hereditary coreflective subcategories of Top A generalization – epireflective subcategories AD-classes and HAD-classes

The same problem can be studied in an epireflective subcategory A

• f Top.

Theorem ([Č2, Theorem 1])

If A is an epireflective subcategory of Top with I2 / ∈ 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

Introduction Heredity of AD-classes References Basic definitions Hereditary coreflective subcategories of Top A generalization – epireflective subcategories 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

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions 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

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions Applications Bireflective subcategories

The operation △

Definition

If X and Y are topological spaces, b ∈ Y and {b} is closed but not

• pen 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 , ga; a ∈ X}, where f : X → X × Y , f (x) = (x, b) and ga : Y → X × Y , ga(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

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions Applications Bireflective subcategories

b Y X a (a, b) y U Vx Figure: The space X △b Y

Martin Sleziak HAD-classes in epireflective subcategories of Top

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions Applications Bireflective subcategories

Let X a

(Y ,b) = the subspace of X△bY 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) → Xa 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 I2 / ∈ 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

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions Applications Bireflective subcategories

b Y X a Xa

(Y,b)

b Y Xa a

The subspace X a

(Y ,b)

The quotient map X a

(Y ,b) → Xa

Martin Sleziak HAD-classes in epireflective subcategories of Top

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions Applications Bireflective subcategories

Sufficient conditions

Let A be an epireflective subcategory of Top with I2 / ∈ 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

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions Applications Bireflective subcategories

Applications

Lemma

Let A be an epireflective subcategory of Top such that I2 / ∈ A. If B = HADA(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 = HADA(B) = ADA(B). Moreover, CH(B) = HCH(B) is hereditary. HADA(D) = HAD-hull of D = the smallest HAD-class in A containing D

Theorem

Let A be an epireflective subcategory of Top such that I2 / ∈ 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

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions Applications Bireflective subcategories

Bireflective subcategories of Top

epireflective subcategories of Top with I2 / ∈ A = the epireflective subcategories of Top which are closed under the formation of prime factors = epireflective subcategories with A ⊆ Top0 epireflective subcategories of Top with I2 ∈ A = bireflective subcategories of Top One-to-one correspondence ([M]): A → A ∩ Top0

Martin Sleziak HAD-classes in epireflective subcategories of Top

Introduction Heredity of AD-classes References Heredity and prime factors The operation △ Sufficient conditions 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