Abstract
We first introduce the notion of a (pre-)apartness between points and subsets in an abstract space X, and derive some elementary properties from our axioms. Each point-set pre-apartness gives rise to a topology–the apartness topology–on X, and to several constructively distinct continuity properties, which are explored in Section 2.3. Limits and the Hausdorff property are discussed in Section 2.4, and product pre-apartness spaces in Section 2.5. In the final section we discuss the role of impredicativity in our theory.
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© 2011 Springer-Verlag Berlin Heidelberg
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Bridges, D.S., Vîţă, L.S. (2011). Point-Set Apartness. In: Apartness and Uniformity. Theory and Applications of Computability. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22415-7_2
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DOI: https://doi.org/10.1007/978-3-642-22415-7_2
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22414-0
Online ISBN: 978-3-642-22415-7
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