Compactification and Closure

Holgate, David

doi:10.2989/16073600009485996

Cited by 4 publications

(6 citation statements)

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“…(b) We observe that it was proved in [17] that if we add to the hypotheses of the above corollary the fact that the subcategory A has a direct reflector R, then A is the compactness class of its associated pullback closure operator P A .…”

Section: Apply To the Galois Connection Cl(x M)mentioning

confidence: 91%

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A categorical approach to absolute closure

Castellini

Holgate

2009

Acta Math Hung

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Section: Apply To the Galois Connection Cl(x M)mentioning

confidence: 91%

“…Now, if X ∈ T Y (P A ) then r X is an isomorphism since it is P A -closed and P A -dense (cf. [17,Theorem 1.6]) and so X ∈ A.…”

Section: -Subobject N I and Finally So Ismentioning

confidence: 96%

A categorical approach to absolute closure

Castellini

Holgate

2009

Acta Math Hung

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“…Compactness and compactification with respect to a categorical closure operator were studied by several authors, in particular, by Clementino et al in [3][4][5] and [6] and by Holgate in [12]. But it was shown in [17] that there is another way of providing a category with a topological structure, namely convergence.…”

Section: Introductionmentioning

confidence: 97%

“…12 Of course, ω has I-indexed concrete products for every index set I. On the other hand, Dir has only finite concrete products (which are defined pointwise).…”

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confidence: 99%

Compactification with Respect to a Generalized-net Convergence on Constructs

Šlapal

2009

Appl Categor Struct

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We introduce and deal with a convergence on (objects of) constructs which is expressed in terms of generalized nets. The generalized nets used are obtained from the usual nets by replacing the construct of directed sets and cofinal maps by an arbitrary construct. Convergence separation and convergence compactness are then introduced in a natural way. We study the convergence compactness and compactification and show that they behave in much the same way as the compactness and compactification of topological spaces.

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“…Nevertheless, the following questions remains open In case of a positive answer to this question, the above intrinsic description extends to all the closure operators of Tych the form C A . It will be nice to find the precise relation of these closure operators to the pullback closure of Holgate [20][21][22].…”

Section: Theorem 69mentioning

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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Compactification and Closure

Cited by 4 publications

References 0 publications

A categorical approach to absolute closure

A categorical approach to absolute closure

Compactification with Respect to a Generalized-net Convergence on Constructs

Conway’s Question: The Chase for Completeness

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