Intrinsic characterizations of C-realcompact spaces

Sudip Kumar Acharyya

India

University of Calcutta

Department of Pure Mathematics

Rakesh Bharati

India

University of Calcutta

Department of Pure Mathematics

Atasi Deb Ray

India

University of Calcutta

Department of Mathematics, Associate Professor.

Submitted: 2020-05-16

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Accepted: 2021-05-10

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Published: 2021-10-01

DOI: https://doi.org/10.4995/agt.2021.13696
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Keywords:

c-realcompact spaces, Banaschewski compactification, c-stable family of closed sets, ideals of closed sets, initially θ-compact spaces

Supporting agencies:

University Grand Commission

New Delhi

research fellowship (F. No. 16-9 (June 2018)/2019 (NET/CSIR))

Abstract:

c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that  X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.
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