Some open categorical problems inTop

Herrlich, Horst; Hušek, Miroslav

doi:10.1007/bf00872983

Cited by 16 publications

(3 citation statements)

References 46 publications

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“…In [7,Problem 7] H. Herrlich and M. Hušek suggest to study coreflective subcategories of Top such that their hereditary coreflective hull is the whole category Top (i.e. SA = Top) and their hereditary coreflective kernel is the subcategory FG.…”

Section: Subcategories Of Top Having Fg As Their Hereditary Coreflectmentioning

confidence: 99%

On Hereditary Coreflective Subcategories of Top

Sleziak¹

2004

Applied Categorical Structures

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Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(A ∪ B) is again FG.

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Section: Subcategories Of Top Having Fg As Their Hereditary Coreflectmentioning

confidence: 99%

On Hereditary Coreflective Subcategories of Top

Sleziak¹

2004

Applied Categorical Structures

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“…Motivated by [11,Problem 7] J. Činčura studied in [3] hereditary coreflective subcategories of the category Top of all topological spaces and continuous maps. He proved a nice characterization of hereditary coreflective subcategories using prime factors of topological spaces.…”

Section: Introductionmentioning

confidence: 99%

Hereditary, Additive and Divisible Classes in Epireflective Subcategories of Top

Sleziak¹

2006

Appl Categor Struct

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Abstract. Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I 2 / ∈ A (here I 2 is the 2-point indiscrete space) were studied in [4]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation of subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A.We show that this is true if A ⊆ Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space.We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [4]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD.

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“…An overview of results can be found in recent survey papers by Herrlich and Hugek ( [10,11]). In particular, the coreflective subconstruct SeqTop of sequential topological spaces mentioned previously, is known to be Cartesian closed and the coreflective subconstruct of all (compact, Hausdorff)-generated topological spaces has been efficiently used by many authors as a nice framework for homotopy theory, topological algebra and duality theory because of its Cartesian closedness.…”

Section: Introductionmentioning

confidence: 99%

On the largest coreflective Cartesian closed subconstruct of Prtop

Lowen-Colebunders

Sonck

1996

Appl Categor Struct

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We show that the subconstruct Fing of Prtop, consisting of all finitely generated pretopological spaces, is the largest Cartesian closed coreflective subeonstmct of Prtop. This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated. Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated. We give a counterexample showing that without finite productivity the previous result does not hold.Mathematics Subject Classifications (1991). 54B30, 18D15, 54A05.

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Some open categorical problems inTop

Cited by 16 publications

References 46 publications

On Hereditary Coreflective Subcategories of Top

On Hereditary Coreflective Subcategories of Top

Hereditary, Additive and Divisible Classes in Epireflective Subcategories of Top

On the largest coreflective Cartesian closed subconstruct of Prtop

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