Abstract
We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.
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Kopperman, R.D., Wilson, R.G. Bitopologies on Totally Ordered Sets. Acta Mathematica Hungarica 84, 159–169 (1999). https://doi.org/10.1023/A:1006615322722
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DOI: https://doi.org/10.1023/A:1006615322722