Ordinal invariants in topology

Kannan, V.

doi:10.1090/memo/0245

Cited by 17 publications

(23 citation statements)

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“…If B is a subcategory of Top, then SB denotes the subcategory of Top consisting of all subspaces of spaces that belong to B. It is well known (see, e.g., [8,1]) that if B is a coreflective subcategory of Top, then SB is also coreflective in Top and, clearly, SB = HC(B).…”

Section: Preliminaries and Notationmentioning

confidence: 99%

Hereditary Coreflective Subcategories of Categories of Topological Spaces

Činčura

2005

Appl Categor Struct

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Let A be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and B be a coreflective subcategory of A. Extending the corresponding result obtained for coreflective subcategories of Top we prove that B is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory B of A containing a non-discrete space is generated by a class of prime spaces and if A is a quotientreflective subcategory of Top, then the assignment B → B ∩ A gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of A that contain FG ∩ A. Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zerodimensional Hausdorff spaces are presented. (2000): 18D15, 54B30. Mathematics Subject Classifications

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Section: Preliminaries and Notationmentioning

confidence: 99%

Hereditary Coreflective Subcategories of Categories of Topological Spaces

Činčura

2005

Appl Categor Struct

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“…On the other hand, the group constructed in Lemma 9 has the property that A = ∪{ A ∩ K : K ∈ K } for any A ⊆ G thus making τ "Fréchet (mod K)" (see [7] for a general discussion of ordinal invariants defined in this manner). So the following question seems natural.…”

Section: An Uncountable Groupmentioning

confidence: 99%

Convergence in topological groups and the Cohen reals

Shibakov

2019

Topology and its Applications

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“…This gives rise to two more closure operators, namely the idempotent hulls Seq ∞ and K ∞ . Let us recall that Seq ∞ is obtained by transfinite ω 1 iterations of Seq, whereas no such common upper bound for all spaces exists for the number of iterations of K in order to get K ∞ [23]. Furthermore, all these closure operators are finer than the Kuratowski closure K, i.e., the C-closure of any subset M is always contained in the usual closure of M. Therefore, in all seven cases, C-dense implies also dense in the usual sense (whereas C-closed subsets need not be closed).…”

Section: Example 32 the Usual Kuratowski Closurementioning

confidence: 99%

Conway’s Question: The Chase for Completeness

Dikranjan

Tarieladze

2007

Appl Categor Struct

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We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone-Čech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Raȋkov completionG. The sequential completeness for The paper was written while the author was visiting the Department of Topology and Geometry of Complutense University of Madrid and was supported by a grant of Grupo Santander (in the program Estancias de Doctores y Tecnólogos Extranjeros en la U.CM.). He takes the opportunity to thank his hosts for the warm hospitality and generous support.The second and the third author acknowledge the financial aid received from MCYT, BFM2003-05878 and FEDER funds.

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Ordinal invariants in topology

Cited by 17 publications

References 0 publications

Hereditary Coreflective Subcategories of Categories of Topological Spaces

Hereditary Coreflective Subcategories of Categories of Topological Spaces

Convergence in topological groups and the Cohen reals

Conway’s Question: The Chase for Completeness

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