Core compactness and diagonality in spaces of open sets

Francisco Jordan

United States

Queensborough Community College

Queensborough Community College, Queens, NY, USA

Frédéric Mynard

United States

Georgia Southern University

Georgia Southern University, Statesboro, GA30460, USA

Submitted: 2013-07-31

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Accepted: 2013-07-31

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DOI: https://doi.org/10.4995/agt.2011.1648
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Keywords:

Scott convergence, Scott topology, Upper Kuratowski convergence, Upper Kuratowski topology, Core compact, Diagonal convergence, Pretopology, Consonance, Infraconsonance

Supporting agencies:

This research was not funded

Abstract:

We investigate when the space OX of open subsets of a topological space X endowed with the Scott topology is core compact. Such conditions turn out to be related to infraconsonance of X, which in turn is characterized in terms of coincidence of the Scott topology of OX × OX with the product of the Scott topologies of OX at (X,X). On the other hand, we characterize diagonality of OX endowed with the Scott convergence and show that this space can be diagonal without being pretopological. New examples are provided to clarify the relationship between pretopologicity, topologicity and diagonality of this important convergence space.
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