Conway's Question: The Chase for Completeness

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 βX (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 ConwC 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 Raı̆kov completion G̃. 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. D. Dikranjan (B) Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy e-mail: [email protected] E. Martín Peinador Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid, Spain e-mail: [email protected] V. Tarieladze N. Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences, Akuri Str. 8, Tbilisi-0193, Georgia e-mail: [email protected] D. Dikranjan, et al. topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.