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An asymmetric approach to filters in strict extensions and quotients

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Abstract

Filters are a fundamental tool in the study of convergence and completeness in topology. On the other hand, downsets have been used extensively for this purpose in the setting of pointfree topology. This paper investigates links between these in the asymmetric context, that is, for biframes and bispaces.

We present an appropriate kind of filter for the asymmetric setting, which we call a bifilter. These form a bispace, functorially so, which proves isomorphic to the spectrum of the downset biframe. As a corollary, downset biframes are seen to be isomorphic to the opens of the bifilter bispace. Both these correspondences are natural isomorphisms.

The join map from a downset biframe to its underlying biframe appears here as a universal strict quotient. We use it to show that the embedding of any T 0 bispace in its bifilter bispace is a universal strict extension.

Banaschewski and Hong [10] have established the importance of general filters for convergence and completeness in the pointfree setting. We conclude this paper with a discussion of an appropriate concept of general bifilter, and show that the right adjoint of the join map mentioned above is a universal general bifilter.

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  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa

    John Frith & Anneliese Schauerte

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  1. John Frith
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  2. Anneliese Schauerte
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Correspondence to Anneliese Schauerte.

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Frith, J., Schauerte, A. An asymmetric approach to filters in strict extensions and quotients. Acta Math Hung 137, 82–103 (2012). https://doi.org/10.1007/s10474-012-0208-5

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