On non-simplicity of topological categories

Hajek, Darrell W.; Mysior, Adam

doi:10.1007/bfb0065261

Cited by 7 publications

(2 citation statements)

References 9 publications

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“…Simplicity problems for TOP and its subcategories defined by means of separation axioms 7o, T\, T^ and by the symmetry axiom Ro have been settled for quite some time [6], [7], [15]. Pretop is known to be simple.…”

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Simplicity of Categories Defined by Symmetry Axioms

Lowen-Colebunders¹,

Szabo²

1991

Can. math. bull.

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We consider two generalizations R0w and R0 of the usual symmetry axiom for topological spaces to arbitrary closure spaces and convergence spaces. It is known that the two properties coincide on Top and define a non-simple subcategory. We show that R0W defines a simple subcategory of closure spaces and R0 a non-simple one. The last negative result follows from the stronger statement that every epireflective subcategory of R0 Conv containing all T1 regular topological spaces is not simple. Similar theorems are shown for the topological categories Fil and Mer.

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Simplicity of Categories Defined by Symmetry Axioms

Lowen-Colebunders¹,

Szabo²

1991

Can. math. bull.

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“…If we take sé = Conv then its bireflective subcategory Top is Conv-simple. Several subcategories of TOP are Conv-simple too, see for instance [4,5,6,7,9,14]. Simplicity remains true if TOP is enlarged to the bireflective subcategory Prtop.…”

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confidence: 99%

On the nonsimplicity of some convergence categories

Lowen¹,

Löwen²

1989

Proc. Amer. Math. Soc.

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Abstract.The category TOP of topological spaces is known to be Convsimple in the sense that there exists a single object E in Top such that Top is the epireflective hull of {£} in the category Conv of convergence spaces. Prtop, the category of pretopological spaces is also Conv-simple. We show that on the contrary the category Pstop of pseudo topological spaces and Conv itself are not Conv-simple. More specifically every epireflective subcategory of Conv which contains all Hausdorff oembedded locally compact spaces is not Conv-simple.For references on topological categories, reflective subcategories and reflective hulls we refer to [6,8,9,10].Let sé be a topological category. All subcategories 38 are assumed to be full and isomorphim-closed. A subcategory 38 of sé is epireflective in sé if it is closed with respect to the formation of products and subobjects in sé . In this context " Y is a subobject of X " means that there exists an embedding from X to Y . This notion coincides with the categorical notion of extremal subobject.Every subcategory < §* of sé is contained in a smallest epireflective subcategory, its epireflective hull, which is denoted by R%. An object A of sé belongs to R% if and only if A is a subobject of a product of objects of <£ . A subcategory 33 of sé is called sé -simple if there exists a single object E of 38 such that 38 is the epireflective hull in sé of the class {E} , i.e., 38 = R{E} .Several examples of this situation are well known. If we take sé = Conv then its bireflective subcategory Top is Conv-simple. Several subcategories of TOP are Conv-simple too, see for instance [4,5,6,7,9,14]. Simplicity remains true if TOP is enlarged to the bireflective subcategory Prtop. This can be derived from results in [1].In this paper we show that simplicity however does not extend to the larger bireflective subcategories Pstop or to Conv itself.

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Categorical Topology — Its Origins, as Exemplified by the Unfolding of the Theory of Topological Reflections and Coreflections before 1971

Herrlich

Strecker

1997

History of Topology

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On non-simplicity of topological categories

Cited by 7 publications

References 9 publications

Simplicity of Categories Defined by Symmetry Axioms

Simplicity of Categories Defined by Symmetry Axioms

On the nonsimplicity of some convergence categories

Categorical Topology — Its Origins, as Exemplified by the Unfolding of the Theory of Topological Reflections and Coreflections before 1971

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